8: Roots and Radicals

  • 8.1: Prelude to Roots and Radicals
  • 8.2: Simplify Expressions with Roots
    • 8.2E: Exercises
  • 8.3: Simplify Radical Expressions
    We will simplify radical expressions in a way similar to how we simplified fractions. A fraction is simplified if there are no common factors in the numerator and denominator. To simplify a fraction, we look for any common factors in the numerator and denominator. A radical expression, ⁿ√ a , is considered simplified if it has no factors of mⁿ. So, to simplify a radical expression, we look for any factors in the radicand that are powers of the index.
    • 8.3E: Exercises
  • 8.4: Simplify Rational Exponents
    Rational exponents are another way of writing expressions with radicals. When we use rational exponents, we can apply the properties of exponents to simplify expressions.
    • 8.4E: Exercises
  • 8.5: Add, Subtract, and Multiply Radical Expressions
    Adding radical expressions with the same index and the same radicand is just like adding like terms. We call radicals with the same index and the same radicand like radicals to remind us they work the same as like terms.
    • 8.5E: Exercises
  • 8.6: Divide Radical Expressions
    We have used the Quotient Property of Radical Expressions to simplify roots of fractions. We will need to use this property ‘in reverse’ to simplify a fraction with radicals. We give the Quotient Property of Radical Expressions again for easy reference. Remember, we assume all variables are greater than or equal to zero so that no absolute value bars re needed.
    • 8.6E: Exercises
  • 8.7: Solve Radical Equations
    • 8.7E: Exercises
  • 8.8: Use Radicals in Functions
    In this section we will extend our previous work with functions to include radicals. If a function is defined by a radical expression, we call it a radical function.
    • 8.8E: Exercises
  • 8.9: Use the Complex Number System
    • 8.9E: Exercises
  • Chapter 8 Review Exercises

Thumbnail: The mathematical expression "The (principal) square root of x". (GPL, David Vignoni (original icon); Flamurai (SVG convertion); bayo (color)).

What are Radical Expressions? (with picture)

A radical expression in algebra is an expression that includes a radical, or root. These are the inverse operations to exponents, or powers. Radical expressions include added roots, multiplied roots and expressions with variables as well as constants. These expressions have three components: the index, the radicand, and the radical. The index is the degree taken, the radicand is the root being derived, and the radical is the symbol itself.

By default, a radical sign symbolizes a square root, but by including different indexes over the radical, cube roots, fourth roots or any whole number root can be taken. Radical expressions can include either numbers or variables under the radical, but the fundamental rules remain the same regardless. To work with radicals, the expressions must be in simplest form this is accomplished by removing factors from the radicand.

The first step in simplifying radicals is breaking the radicand into the factors needed to equal the number. Then, any perfect square factors must be placed to the left of the radical. For example, √ 45 can be expressed as √ 9*5 , or 3√ 5 .

To add radical expressions, the index and radicand must be the same. After these two requirements have been met, the numbers outside the radical can be added or subtracted. If the radicals cannot be simplified, the expression has to remain in unlike form. For example, √ 2 +√ 5 cannot be simplified because there are no factors to separate. Both terms are in their simplest form.

Multiplying and dividing radical expressions works using the same rules. Products and quotients of radical expressions with like indexes and radicands can be expressed under a single radical. The distributive property works in the same fashion as it does with integer expressions: a(b+c)=ab+ac. The number outside the parenthesis should be multiplied by each term inside parenthesis in turn, retaining addition and subtraction operations. After all terms inside the distributive parentheses are multiplied, the radicals have to be simplified as usual.

Radical expressions that are part of an equation are solved by eliminating the radicals according to the index. Normal radicals are eliminated by squaring therefore, both sides of the equation are squared. For example, the equation √ x =15 is solved by squaring the square root of x on one side of the equation and 15 on the right, yielding a result of 225.

8: Roots and Radicals

Here we will define, analyze, simplify, and calculate the square root of 8. We start off with the definition and then answer some common questions about the square root of 8. Then, we will show you different ways of calculating the square root of 8 with and without a computer or calculator. We have a lot of information to share, so let's get started!

Square root of 8 definition
The square root of 8 in mathematical form is written with the radical sign like this √8. We call this the square root of 8 in radical form. The square root of 8 is a quantity (q) that when multiplied by itself will equal 8.

Is 8 a perfect square?
8 is a perfect square if the square root of 8 equals a whole number. As we have calculated further down on this page, the square root of 8 is not a whole number.

8 is not a perfect square.

Is the square root of 8 rational or irrational?
The square root of 8 is a rational number if 8 is a perfect square. It is an irrational number if it is not a perfect square. Since 8 is not a perfect square, it is an irrational number. This means that the answer to "the square root of 8?" will have an infinite number of decimals. The decimals will not terminate and you cannot make it into an exact fraction.

√ 8 is an irrational number

Can the square root of 8 be simplified?
You can simplify 8 if you can make 8 inside the radical smaller. We call this process "to simplify a surd". The square root of 8 can be simplified.

How to calculate the square root of 8 with a calculator
The easiest and most boring way to calculate the square root of 8 is to use your calculator! Simply type in 8 followed by √x to get the answer. We did that with our calculator and got the following answer with 9 decimal numbers:

How to calculate the square root of 8 with a computer
If you are using a computer that has Excel or Numbers, then you can enter SQRT(8) in a cell to get the square root of 8. Below is the result we got with 13 decimals. We call this the square root of 8 in decimal form.

What is the square root of 8 rounded?
The square root of 8 rounded to the nearest tenth, means that you want one digit after the decimal point. The square root of 8 rounded to the nearest hundredth, means that you want two digits after the decimal point. The square root of 8 rounded to the nearest thousandth, means that you want three digits after the decimal point.

What is the square root of 8 as a fraction?
Like we said above, since the square root of 8 is an irrational number, we cannot make it into an exact fraction. However, we can make it into an approximate fraction using the square root of 8 rounded to the nearest hundredth.

√ 8
≈ 2.83/1
≈ 283/100
≈ 2 83/100

What is the square root of 8 written with an exponent?
All square roots can be converted to a number (base) with a fractional exponent. The square root of 8 is no exception. Here is the rule and the answer to "the square root of 8 converted to a base with an exponent?":

How to find the square root of 8 by long division method
Here we will show you how to calculate the square root of 8 using the long division method with one decimal place accuracy. This is the lost art of how they calculated the square root of 8 by hand before modern technology was invented.

Step 1)
Set up 8 in pairs of two digits from right to left and attach one set of 00 because we want one decimal:


Step 2)
Starting with the first set: the largest perfect square less than or equal to 8 is 4, and the square root of 4 is 2. Therefore, put 2 on top and 4 at the bottom like this:


Step 3)
Calculate 8 minus 4 and put the difference below. Then move down the next set of numbers.


Step 4)
Double the number in green on top: 2 × 2 = 4. Then, use 4 and the bottom number to make this problem:

The question marks are "blank" and the same "blank". With trial and error, we found the largest number "blank" can be is 8. Now, enter 8 on top:


That's it! The answer is on top. The square root of 8 with one digit decimal accuracy is 2.8.

Square Root of a Number
Please enter another number in the box below to get the square root of the number and other detailed information like you got for 8 on this page.

Remember that negative times negative equals positive. Thus, the square root of 8 does not only have the positive answer that we have explained above, but also the negative counterpart.

We often refer to perfect square roots on this page. You may want to use the list of perfect squares for reference.

Square Root of 9
Here is the next number on our list that we have equally detailed square root information about.

What is the principal cubic root of $-8$?

it seems that WolframAlpha consider as principal root the root with the minimum value of the argument, i.e $e^$, but usually, if there is a real root this is considered the principal root.

Mathematicians (as opposed to elementary textbook writers) consider complex numbers. To make the principal value of the cube root continuous in as large a region of the complex plane as possible, and to have it agree with the conventional cube root for positive numbers, they choose the cube root with argument in $(-pi/3 , pi/3]$. This fits together with a systematic choice of principal value for logarithm, arctangent, and others.

There are three complex cubic roots of $-8$.

Namely $2 e^$, $2 e^$, $2 e^ $. You get this using the polar form(s) of $-8$, that is $8 e^$ and more generally $8 e^$ for $k $ and integer.

The middle one is $-2$ and the first one is what Wolfram Alpha gives.

The rational for W|A making that choice is that it considers as principal root the one where the principal argument of $-8$, that is $pi$, is divided by $3$.

The rational of your book should be that they want to be cubic roots of reals to be real.

It is a matter of definition/convention. In some sense I prefer the former, but would use the latter in the right context too.

In any case, you should follow your book or talk to your instructor about it, as otherwise there will be confusion.

The Politics of a Second Gilded Age

Mother’s Day is known to most Americans as a day when you should call your mom. And you probably should! But it also has a rich political history.

Mother’s Day began in 1858 when Ann Jarvis, an Appalachian housewife and mother to at least eleven children, organized “Mother’s Work Days” to improve sanitation, in a time when polluted water and disease-bearing pests were major causes of death in poor communities like hers. Jarvis was also a peace activist who organized Mother’s Day Work Clubs to care for soldiers on both sides of the Civil War.

When Ann Jarvis died in 1905, her daughter, Anna Jarvis, campaigned for an official Mother’s Day to honor her own mother’s lifelong activism. In 1914, her efforts succeeded: Congress passed a resolution making Mother’s Day official.

A portrait of Ann Jarvis on the program for the first official Mother’s Day service, in May 1908. (West Virginia and Regional History Center, WVU Libraries)

The holiday also has an antiwar history. In 1872, after the brutal Franco-Prussian war, Julia Ward Howe, a peace activist who wrote “Battle Hymn of the Republic,” established a day for mothers and antiwar protest. As my Jacobin colleague Branko Marketic wrote in 2019, her vision was an internationalist one, calling for a “general congress of women without limit of nationality” to “promote the alliance of the different nationalities, the amicable settlement of international questions, the great and general interests of peace.” Howe organized protests and political conferences on the day, also calling it a “Women’s Peace Festival,” for many years.

For much of the twentieth century, the political ideals of Ann Jarvis and Julia Ward Howe were eclipsed as Mother’s Day became associated with treacly Hallmark commercialism. Anna Jarvis herself became disgusted with it by the early 1920s, even boycotting florists and picketing a confectioners’ convention to protest Mother’s Day price-gouging on flowers and sweets.

But in the 1950s and ’60s, Women Strike for Peace (WSP) held Mother’s Day actions around the country to protest the life-threatening nuclear arms race between the United States and the Soviet Union. The group had a respectable, middle-class image, often protesting while wearing white gloves and pushing strollers. The anti-Communist House Un-American Activities Committee (HUAC), however, took a great interest in WSP because of the many Communists involved in the group or closely associated with it. Indeed, a HUAC report on the group noted that, when communists use the word “peace,” it may sound wholesome like mom and apple pie, but a critique of capitalism is implied. (The committee was correct.)

Activists from Women Strike for Peace holding placards relating to the Cuban missile crisis in 1962. (Library of Congress)

Another group known for Mother’s Day actions was the National Welfare Rights Organization (NWRO), made up of poor women demanding the economic right to survival and often much more. As sociologist Wilson Sherwin has found, welfare rights activists often had a radical critique of wage labor and a vision of a society in which everyone was entitled to comfort, leisure, and pleasure, whether they worked for a capitalist boss or not.

The NWRO’s most famous Mother’s Day action was a march on Washington on May 12, 1968, led by Coretta Scott King (who was also active in WSP) just after the assassination of Martin Luther King Jr. Organized with the Southern Christian Leadership Conference, the march had been intended to launch King’s Poor People’s Campaign (conceived just before his murder) to demand more government support for the poor, but the NWRO also used the occasion to draw attention to its effort to repeal the 1967 Social Security Amendments, which added work incentives to the program. In addition to a march through Washington, DC, the NWRO also held smaller actions around the country. Members of WSP joined them.

National Welfare Rights Organization activists marching in Washington, DC, May 1968. (Jack Rottier Collection / George Mason University Libraries)

So Mother’s Day has radical, materialist roots and history. Still, as socialist feminist scholar Kristen Ghodsee observes, socialists have often rejected Mother’s Day, preferring to celebrate and protest on International Women’s Day, March 8, which celebrates women as workers, a broader category that can include the labor of caring for their own children at home. (Indeed, in Albania and some other formerly communist Eastern European countries, Mother’s Day is celebrated only as part of International Women’s Day.) In favoring March 8 over a mother-exclusive day, Ghodsee explains, “Socialists embraced the radical idea that a woman is more than a mother, though she may be that, too.”

This is a fair point. The WSP and NWRO shrewdly used Mother’s Day — and, as historian Georgina Denton has noted, the moral authority of motherhood — to bring radical ideas into the mainstream. They sometimes used essentialist language that radicals would likely reject today, for some of the reasons Ghodsee suggests. At the 1968 march, Coretta Scott King, emphasizing “mother power,” discussed the political responsibilities of mothers:

Since women have been entrusted with the sacred task of giving birth and rearing children, transmitting the values and cultural heritage of the nation, we have a special commission at this time to nurture, protect, and save these lives from destruction.

King, of course, was using traditionalist rhetoric in the service of radical and liberationist politics. But in some countries, the politics of Mother’s Day have been openly conservative and natalist.

Code Pink activists in front of the White House. (Code Pink)

For example, in France, Mother’s Day began in the context of elite alarm over low birth rates, as a way of honoring mothers of large families. (The French government still gives out medals to such reproductive overachievers.) In Germany, too, Mother’s Day originates in natalist and nationalist politics. Intended to boost birth rates and promote the production of more Aryans, it became an official holiday under the Nazis.

That lends some serious ammunition to the Mother’s Day haters.

Still, in the United States, Mother’s Day comes out of a rich history of protest — one that is worth reviving.

In our own time, Code Pink, a feminist antiwar group drawing on Julia Ward Howe’s legacy, sometimes holds Mother’s Day protests of US foreign policy. This aspect of the Mother’s Day tradition will be relevant for as long as our government continues to wage endless imperialist wars. The NWRO’s demand for more government support for mothers feels equally relevant.

So consider that legacy as you celebrate Mother’s Day today. Also, call your mom.

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Square Roots and Radicals

A square root is defined as a number which when multiplied by itself gives a real
non-negative number called a square.

A square root is best defined using geometry where, considering a square (which
is a four sided polygon whose sides are all equal), a square root is defined as
the length of the diagonal of this square (a diagonal is a line drawn from one vertex/corner
to the opposite vertex of the square).

A radical is a root of a number. A square root is a radical. Roots can be square
roots, cube roots, fourth roots and so on.

A square root is commonly shown as

is known as the radical sign and

A square root of a number can also be represented as

where we say that in the above, we’re finding the nth root of x. For
more on the above notation, refer to section on

A radical can also be represented as

A square root is also represented as

Every square has two square roots one positive and the other negative. This is
shown as:

This can be proved in the following way. Consider a number, a

the latter is because a negative multiplied by a negative equals a positive.

Thus it follows that any real positive number has two roots. But when talking about

only refers to +x which is known as the principal square root. So despite
having said above that

especially if the
is used.

But if the question asked is in the form

always give both the positive and negative roots, i.e.

Although any real positive number can be considered a square number and thus has
a square root, we only consider numbers with whole number square roots as squares.

8: Roots and Radicals

You may want to review the sections on integer exponents and exponents in general before you read this section.

Radical notation

  • The index, n , must be a positive integer.
  • The above radical is spoken as &ldquothe n th root of b &rdquo, for any index except n = 2 when we say &ldquothe square root of b &rdquo and n = 3 when we say &ldquothe cube root of b &rdquo.
  • An index of 2, for the square root, is usually not written.
  • The words &ldquoradical&rdquo and &ldquoroot&rdquo will be used interchangably in this section.
  • The roots in this section have (almost) nothing to do with roots of an equation.

This means that for every property or rule that holds for an exponential there is a corresponding property or rule for a radical.

Root of a power and power of a root

Thus the n th power and the n th root are inverse functions.

Root of a product or quotient

Simplest form of a radical

  • The radicand has been reduced as much as possible. (See the first example above.) This is done by removing factors from the radical.

Removing factors from the radicand

Suppose that the index of the radical is n . Then factor the radicand so that one or more of the factors is a perfect n th power. Then rewrite the root of the product as a product of roots and use the fact that to simplify those factors. This process is called removing factors from the radicand .

Rationalizing the denominator

An expression is considered to be simpler when its denominator contains no radicals.

Suppose that the denominator of a fraction contains a square root. Then multiply both the numerator and denominator of the fraction by that square root and simplify. This may produce a radical in the numerator but it will eliminate the radical from the denominator. This process is called rationalizing the denominator .

Example: Simplify the expression .

  • Multiply the numerator and denominator by the square root of 5 (shown below in blue).
  • Multiply the fractions.
  • Use the fact that . The result is a radical in the numerator but none in the denominator.

Suppose that a square root contains a fraction. Then multiply both the numerator and denominator of the fraction by the denominator of the fraction and simplify. This may produce a radical in the numerator but it will eliminate the radical from the denominator. This process is also called rationalizing the denominator .

Example: Simplify the expression .

  • The square root contains a fraction and the denominator of the fraction is 5 y . Multiply the numerator and denominator of the fraction by 5 y (shown below in blue).
  • Multiply the fractions.
  • Convert the root of a quotient to a quotient of roots.
  • Use the fact that . The result is a radical remaining in the numerator but none in the denominator.

Combining products of radicals

We saw above that the root of a product could be rewritten as a product of roots. Here we want to go the other way. It sometimes happens that converting a product of roots to the root of a product produces a perfect square factor and that factor can then be removed from the radicand.

Example: Simplify the expression .

Solution: Write this as the root of a product, namely of 60, and then notice that 60 has a perfect square factor of 4, which can then be removed from the radicand.

Reducing the index

It is sometimes possible to reduce the index by writing the radical in exponential form and then reducing the fractional exponent to lowest terms.

Example: Simplify the expression .

Solution: Write this in exponential notation and use the exponentiation property of exponents to give the single exponent 2/6 and then reduce the fraction 2/6 to 1/3.

Binomial denominators containing radicals

Suppose that the denominator of a fraction is a binomial (i.e. it contains two terms) and that one or both of those terms is a radical. Then multiplying the numerator and denominator of the fraction by the binomial conjugate of the denominator and distributing will eliminate all radicals from the denominator. (Note that the binomial conjugate of the binomial a + b is the binomial a &minus b and vice versa .

Example: Eliminate the radical from the denominator of the expression .

  • Multiply the numerator and denominator by the binomial conjugate of the denominator (shown below in blue).
  • Multiply the fractions.
  • Distribute in the numerator. In general the radicals survive. Distribute in the denominator. The cross-terms cancel so that the radicals disappear, which is the whole point of using the binomial conjugate.

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Down To The Roots: The Radical Politics of Reggae

Bob Marley is a paradox. Part “Is This Love,” part “Buffalo Soldier,” Bob Marley is the poster boy and evangelist for reggae music, viewed through a commodified 21st-century lens that sanitizes the truly radical nature of his music and legacy. Although less discussed among much of Marley’s contemporary worldwide fanbase, the singer-guitarist’s political messaging and activism speaks to the generations-long resonance and power of reggae music.

“For a lot of people in the 21st century, Bob Marley is all about ‘One Love,’ and he’s been systematically defanged. But for a lot of people, it’s still about ‘Get Up, Stand Up,’” said Don Letts, a first-generation Black British man of Jamaican descent whose contributions to reggae and punk include multiple documentaries, videos for The Clash, BBC 6 Music shows, and the Reggae 45s podcast co-presented with Turtle Bay. This dichotomy is certainly present in Marley’s work and reverberates through UNESCO’s recent designation of reggae as an “intangible cultural heritage.”

Despite (or, perhaps, because of) its feel-good, island vibe and emphasis on bass and rhythm, reggae is a unique vehicle for delivering messages of empowerment, Afrocentrism, and anti-colonial critique. It’s also inherently political music, grown from the freedom sounds of Jamaican independence in 1962 and blossoming into the heavier, socially conscious sound that reflected less celebratory times in 1968. Reggae – as well as its sonic predecessor’s ska and rocksteady – spread like wildfire throughout the island, holding up a groovy mirror to Jamaican national politics and the state of marginalized people the world over.

While Marley started out singing upbeat covers of “What’s New Pussycat” with The Wailers in 1965, his later discography includes albums such as Rebel Music and Uprising! Marley’s breakout reggae-rock crossover album, 1973’s Catch a Fire, features a number of heavy political themes that are exemplary of the consciousness-raising power of reggae. “Slave Driver” makes a direct connection between the slave trade and continued inequity among Black people: “Every time I hear the crack of a whip/My blood runs cold/I remember on the slave ship/How they brutalize the very souls/Today they say that we are free/Only to be chained in poverty/Good God, I think it's illiteracy/It's only a machine that makes money.”

Before the public acknowledgment of the legacy of colonialism, songs like “Slave Driver” served as a vehicle for important, alternative narratives. “The music has informed the whole planet it’s become part of the fabric of popular music, and some people picked up on the roots radical, get up stand up reportage,” Letts told The North Star, noting his own reggae-fueled consciousness in the early ‘70s. “In the UK they’re showing images of white Jesus and perfection, and I could never achieve that, and nor did I want to. And here comes reggae and they’re talking about Africa and not being so Eurocentric in our ideology. It taught me about my own roots and what we had to bring to the party.”

Everything from Marley’s “African Herbsman,” to Burning Spear’s “Door Peep Shall Not Enter,” and The Abyssinians’ "Satta Massagana" subverted a colonial mentality which was still present after independence, said Carter Van Pelt, VP Records’ director of catalog development and founder of Coney Island Reggae on The Boardwalk. “After the colonial masters are no longer in power you have a system that, no matter who the person is, is perpetuating aspects of domination and subjugation that are coming from slavery. Naturally, there will be an oppressed part of society that resists, and reggae articulated that.”

This aural education predates other ‘70s era protest music and certainly informed the messaging of hip hop and punk. The first wave of reggae is entirely rooted in the Jamaican experience and the nation’s long history of dissent and protest, though it’s not the first popular island sound to question the status quo. Rooted in the traditions of enslaved West Africans brought to the Caribbean, calypso long predated reggae and was a method of “communicating and interpreting political events, and a primary news source for many islanders,” according to the BBC. In the early ‘60s, rude boys — a youth subculture known for their sharp dress, attitude, and violent tendencies — made international headlines for the way they challenged the system with a cool and copacetic attitude. Both ska and early reggae are associated with rude boys Bob Marley sang in praise of and against the rude boy, and “rudies” would appear in countless singles throughout the decade and two songs in 1979.

1968, the year reggae was born, was volatile and creatively rich. Jamaica experienced widespread public-sector workers’ strikes for better wages prominent Guyanese professor and Black Power advocate Walter Rodney was banned from returning to Jamaica and labeled a communist threat, Patricia Meschino wrote in Billboard. The resulting Rodney Riots in Kingston resulted in six deaths, nearly 100 arrests, and damages to property. That same year, The Maytals released “Do The Reggay” and a slew of singles using the word “reggae” followed.

Others engaged in the much loved, inherently subversive Rastafari belief system, which emerged in Jamaica in the 1930s and embraced an Afrocentric view that encouraged followers to escape Babylon — ostensibly the corrupt, colonial world. In 1966, Ethiopian emperor Haile Selassie’s (whom the Rastafari view as divine) visited the island, striking a chord in the newly independent nation and the 12 Tribes of Israel was founded two years later. Bob Marley was a member of this branch of Rastafarianism, and the religion’s messages permeate throughout the history of reggae music.

“A lot of political messages were being articulated by people who were Rasta, or if they weren’t they were very sympathetic to the cause of Rastafari and the way Rastafari viewed the relationship with Western civilization,” Van Pelt said, adding that Rasta culture also preached a message of unity which broadened the audience for reggae. “Maybe that was promoted as a way to make the music less intimidating. And a lot of it just has a positive spirit.”

Journalist and NYU professor Vivien Goldman has made a career of reggae and worked with Bob Marley as a PR officer at Island Records. “[Bob Marley] was really spiritually committed, and believed in Jah and believed in reggae,” Goldman told The North Star. “He said to me, ‘This is the music the Bible speaks of in so many places.’ For him…music was the weapon spiritually and socially.”

As Bob Marley and his generation reckoned with what it meant to be in control of their destiny as the “Sons of Slaves,” reggae’s messages spread to Black people (and woke others) throughout the world who iterated on its sounds. Punk is among the genres influenced by reggae (and heard throughout songs by The Clash, Bad Brains, and even Elvis Costello), and this cross-pollination gave birth to a punky reggae hybrid called Two Tone. This short-lived, second wave of ska grabbed hold in the UK in the late ‘70s with a mixture of political sensibility and upbeat rhythm and was popularized by multi-ethnic bands such as The Selecter and The Specials, whose single “Ghost Town” practically defined England in ‘79. It was uncommon to see Black and white people together on stage in England at the time — which was in the throes of racist violence and fear mongering from fascist political group the National Front, and its Black population terrorized by equally racist “sus” laws — and Two Tone bridged a racial divide.

Around the same time, dub reggae and soundsystem culture (mobile, underground parties featuring homemade stereo equipment and emcees) created community among its mostly Black, often marginalized fans. This culture was exquisitely captured in the incendiary 1980 film Babylon, which officially premiered in the US March 8, and continues today in multiple countries. Both Two Tone and the dub soundsystem scene were influenced by an earlier generation of white youth and West Indian immigrants, who found commonality in their working-class roots and love of ska and rocksteady music such as Desmond Dekker’s “Israelites” – a phenom described in detail in the Trojan Records documentary Rude Boy which also continues worldwide.

Back on the island, an eight-year experiment in socialism that began in 1972 under the People’s National Party (PNP) government brought education, health care, and labor rights. At the same time, poor economic conditions, politically-affiliated gang violence, and organized crime, as well as an alliance with communist Cuba, put the PNP in opposition to the American government and made it a target of the CIA. An assassination attempt was made on Bob Marley in 1976, ahead of the Smile Jamaica peace concert which some saw as supportive of the PNP. Multiple people were wounded, and Marley was shot in the chest and arm, while his wife, Rita, was shot in the head. Miraculously, no one died, and The Wailers played the event. Several years later, the US threw its power behind the Jamaica Labour Party (JLP) in the 1980 election of Edward Seaga, who overturned much of the PNP’s socialist programs. “Things improved somewhat economically, so I think that took a lot of the wind out of the sails of reggae,” Van Pelt noted. Bob Marley would pass away in 1981 from cancer.

Fifty years after reggae came to wake the town, its political fury has calmed down. Letts and Van Pelt surmise that the consumerism of the 1980s, years of political turmoil in Jamaica, and trends of hypersexualized dancehall and instrumental dub all but snuffed out the flames of golden age reggae (often characterized as “roots reggae”). “Rastafari lyrics are alive and well, but they don’t point toward anything specific. It’s the abstract of chanting down Babylon,” Van Pelt said, adding modern artists such as Chronixx, Kabaka Pyramid, Protégé, Jesse Royal, and Jah Nine are considered part of a reggae revival that’s Afrocentric but not expressly political. “There was a time when it felt like music as a tool for social change was out of vogue. There are still people who are inspired and understand the possibility of music to shake things up, but no one is going to put them on a pedestal,” Letts adds.

Goldman points to Tanya Stephens as a modern reggae artist tackling social issues such as homophobia. Still, “it’s arguable how much of the revolutionary content really stayed in foreground today. Has the let’s roll a fattie brigade and sort of veg out taken over,” Goldman questioned. “We’re hoping some echoes of it still remain to stir people up on a different level and use the music to help achieve the truth and rights that were so essential in the first wave.”

However, the fact that reggae sounds have informed the planet is a good thing, even if it’s in the form of easy listening. “It’s found a home in both camps. For those who were tuned into the rebellious aspect, there’s that, then there’s other people who picked up on the good time skanking music that makes you want to dance,” Letts said. “[Bob Marley] wasn’t one dimensional. He wasn’t just One Love, just as he wasn’t Get Up, Stand Up. Bob was inspired by Black Panthers he was a rebel at heart but knew you can’t be fighting all the time. He was that raggamuffin rude boy and a spiritual lover.”

The continued political resonance of reggae follows much debate about UNESCO’s affirmation of the genre’s cultural importance. “Jamaicans do not need the UN to endorse the soundtrack of their lives,” Dotun Adebayo wrote in The Guardian. “It’s not the Jamaican people but the Jamaican establishment that needs to hear that. The very same Jamaican establishment that fought against reggae for years until they realized it was bringing in more revenue than the nation’s ailing bauxite industry so they had to incorporate it.”

Letts said the UNESCO nod seemed condescending. “There seems to be a lot of money going into the island through music and tourism, but it doesn’t seem that it trickles down to the man on the street” — an inequity reggae fought against in the 1970s. Van Pelt noted that Jamaica’s minister of culture, who spearheaded the UNESCO designation, came up with reggae. “I think it helps everyone who’s involved in reggae, who take it serious as an art form. What it will do to change the way the Jamaican government treats the music, we have yet to see.”

The real political legacy of reggae is not a crowing swipe of the pen by UNESCO it’s the way the music has disseminated throughout the world. Letts noted that there are hundreds of dub sound systems in faraway places like Croatia. “It’s a very interesting phenomenon, which is so far removed from their own culture. There’s something about Black culture speaking to the souls of these white kids, and you hope that it resonates and helps facilitate us coming together.”

The fact that people in the far corners of the globe have Bob Marley posters on their walls, know of the Rastafari movement, has created hundreds of roots reggae bands, and sing songs of revolution and unity is a testament to the ongoing power of reggae. Its longevity is inherently political. “We’re talking about an island that spent 100 years under the yoke of colonization, and then in the 20th century, it colonized the whole world,” Letts said.

8: Roots and Radicals

Here are the steps required for Solving Problems Containing a Radical:

Step 1: Isolate the radical by moving all other terms to the other side of the equation.
Step 2: Look at the index of the radical. Raise each side of the equation to the power of the index. For example, if the radical is a cube root, then the index is 3 and we need to cube each side of the equation.
Step 3: Solve the equation found in step 2. This step may require distributing (or FOILing), combining like terms, isolating the variable, or solving by factoring depending on the remaining terms.
Step 4: Check your answer, if you need to. If the index is even, such as a square root or fourth root, then we must check your answer(s) by plugging the answer into the original equation to see if it correct. If the index is odd, such as a cube root or fifth root, then there is no need to check the answer to see if it is correct. Only even indices can produce incorrect answer.

Example 1 &ndash Solve:

Example 2 &ndashSolve:

Example 3 &ndashSolve:

Example 4 &ndashSolve:

If the problem contains rational (or fractional) powers rather then radicals, just rewrite the problem using radicals rather then rational (or fractional) exponents and solve the problem using the four steps shown above.

8: Roots and Radicals

Dateline: AmeriKKKa, 1968 -- 2006

WHEN I WAS VERY YOUNG, majoring in marijuana at the university, hanging out with the Progressive Labor Party, and skipping through the clouds of tear gas on Telegraph Avenue in Berkeley, I was convinced that any war that would send my long-haired, sensitive, poetic and acid-tripping self off to wade through rice paddies in Vietnam just had to be wrong, wrong, wrong .

In those years it was easy to see the United States through red-tinted glasses. All you had to do was load a Chillum , roll another Giant Doobie, put "Blonde on Blonde" on the turntable, plug in the Bongomatic and light everything up. Like so many others in that long ago land of Nod-Out, this ritual was my major course of study.

Once this gentle ritual sufficiently soothed my tortured soul I'd often make my way (s l o w l y) to the daily Vietnam Day Committee meeting for a righteous rap session on how "the man can't bust our music or our movement." Then I'd float my way back home to listen to my hot red-diaper girlfriend rhapsodize about her Worker's Party parents and natter on about old Progressive Labor Party parties in New York that seemed to center not on politics but on heroin suppositories. She thought "those were the days."

I wasn't so sure, but she had cool Communist credentials signed off on by no less than the dowager princess of the American Communist Party Bettina Aptheker, so I was inclined to go along with her drivel in order to get along with her. Living with a red-diaper princess who was on the steering committee of the VDC was, in those days at Berkeley, better than going steady with the Homecoming Queen.

In later life, my princess was due to come down in the world. The last I looked she was counseling families into or out of the family state. Just another ordinary therapist wading through the muck of Urban angst. Back then she was professionally oppressed by the fascist war machine and so was I. So was every other college-deferment clutching coward of my time. Fear and lust controlled us well. She, and so many others, "said 'Yes!' to boys who said 'No!'" so I memorized all the ways in which we were oppressed. She was always giving a test on this subject and I didn't do so bad.

I also found that, like any good Berkeley radical, you needed -- in this realm of unremitting oppressions so thick and so multiple that counting was foolish -- to find some good friends some very good friends both at home and abroad. And so you looked around, not so much for friends, but for enemies of your enemy, the oppressive AmeriKKKa. You looked around the world using the dubious intellectual filter: "The enemy of my enemy is my friend."

When you ran everything you said and did through that filter you had no end of friends in the 60s and early 70s. Some of them even had guns, ammo, armies, armor and nuclear weapons.

The "friends" of our slice of the young America included, but were not limited to, The Soviet Union, The People's Republic of China, Castro's Cuba, and socialist and communist parties stretching across Europe and down through Mexico, Central and South America. Elsewhere they included the Viet Minh, the Viet Cong, the Khmer Rouge, the Red Army Faction. on and on until the international litany of political dementia girdled the globe in a seamless political landscape of perpetual Revolution, sexual and otherwise. You and your oppressed and draftable friends saw those groups and nations as "righteous." You saw them not as the totalitarian serial killers they were, but as the secular saviors of civilization. They were, well, just very cool guys. They were "happening, bro."

When you got done with feeling cool about those friendly states and organizations, you still weren't out of friends. Instead, you just went on to the "enemies of your enemy" that were not necessarily rooted in real estate, but in the mind and the culture. These groups have been summed up in a stunning fashion by Paul Mann in his perceptive essay "Stupid Undergrounds." They were --take a deep breath --

As you can see we had plenty of friends.

And they and we all grew older. We survived and thrived. Some even grew up, but only a few. For most of us -- no matter what was our lot in later life -- it would always be 1968. We so loved being "The Lost Boys."

Although I was of -- and among -- many of the groups above, none of them are among my friends any longer. I have, alas, far fewer friends. Indeed, as my strange political odyssey of saying "Goodbye to all that" continues, old friends seem to melt away like the highland mist at high noon in the desert. It is sad, but still, with friends like those.

Over the four decades since 1968, the list of regimes dedicated to, and capable of, the destruction of the United States shrank. They either took a long dirt nap in history, or are now shambling towards the graveyard of all other failed but deadly fascist ideologies. The political genius and destiny of the United States lies, after all, in the fact that we do not require you to be a friend. You simply have to not be an enemy.

The American Way is, after all, that nothing need be personal when it can just be business. One on one, Americans can be very warm, understanding and generous. But piss us off too much and we'll bomb your cities to rubble. We don't like business to be disrupted too much.

In all this, the world at large has gone forward and, all in all, improved for the better. We call it "Globalization" and it seems, slowly, to be working out well for most of those people who have, as they say, "gotten with the program."

But there remains a residual group of Americans who, although they batten off the program, don't want to get with it at all -- except when it comes time to buy a new Prius, vacation in Provance, or score a country home. They take pride in never having sold out, even as they buy in. They are "the Not-So-Great Generation."

Those Americans of the 60s whose fantasies were lit by a dream of a destroyed United States have very few friends left out of the long list of countries once dedicated to totalitarianism. And the list becomes shorter with each passing year.

Time and chance also makes the list of those Americans still dedicated to becoming life-long friends of countries and movements dedicated to the destruction of AmeriKKKa shorter every year. Yet most strangely still live and thrive in the very country they hate the most.

These dreary souls without a country have made prosperous lives for themselves in our local, state and national governments and politics. They are legion in the comfortable realms of academia, entertainment world, and the media. Graying they tint their hair and continue in their quest for an enemy of their enemy to make their friend. Balding, they tie what they have back in a ponytail and strut on impertube.

They are the American Left and, risen from their impoverished conditions in 1968, they now have tenure, high position, or acolytes from which they draw comfortable stipends. Of late, they've taken more and more to coffee klatches with Islamic fundamentalists who, if they don't have the armies to bring about the destruction of the United States, have at least shown they've got enough hate to kill Americans here and abroad retail and wholesale. Besides, they're out shopping for a nuclear weapon and some smallpox, so what's not to like about these guys from an American Leftist's point of view?

Sadly the longer list of Paul Mann's Stupid Undergrounds shows no signs of shortening. Fueled by the vapid culture of cool it gets longer by the day. And it is from within this expanding list of Stupid Undergrounds that the American Left of today draws not only its strength but its fresh and much younger converts.

If it were only the denizens of these fringe groups that supplied the ideological cannon fodder of the American Left, it would be a small matter to marginalize them since their very mindsets marginalize them from the square numbered "1." Indeed, just a few years ago, they could only exist within the rarefied environment of on-campus humanities and ethnic-studies departments. Once removed from these hyperbaric chambers, their failure to thrive in the world outside -- absent a position in various media companies and Washington Wonk Tanks -- was assured. They were, if not really useful idiots, harmless idiots.

Sadly that is no longer the case. Recently a very large and significant American institution has stripped to the buff, oiled up, and made its body politic freely available to the tender mercies and tough love of the American Left. Indeed, the capture of this group is the single significant achievement of the American Left in decades. With the elevation of Howard Dean, the canonization of Hillary Clinton, the sanctification of Ted Kennedy, the renovation of Nancy Pelosi, the self-defenestration of Barbara Boxer, and the deification of Barack H. Obama, it is clear that the political base of the American Left has now migrated from the fringes of our political arena to the dead center of the Democratic Party. And it is there to stay.

The American Left now controls the political party that calls upon the allegiance of nearly half of the country. It is the political party that is the Plantation Party of African-Americans. It is a party that holds its members now not with the plans of what it will do for them in the future, but with the fading memories of what it did for them in the past when it was a great and honorable party. The American Left will remain in control of this Party's shell since it has brought with it not only its failed ideology and all the rag-tag constituents of the Stupid Undergrounds of America, but the very fuel source of these groups itself -- Bush Hate. And on the Left today, Bush-Hate, more than money, is the new mother's milk of our darkening politics. With Bush-Hate money can always be had. Throw that out of the Democratic Party and what money there is currently coming into the party will surely flow away.

It is true that Bush-Hate brings with it a number of disturbing ideological contradictions to the center of the Democratic Party. These were best summed up recently by Pamela Bone in her essay The Silence of the Feminists :

The conquest of the Democratic Party by the American Left which has now been consummated and will shortly be consolidated is, of course, bad news for the Democrats and for the country as a whole. A vital two or even three party system is essential to the long term balance of the Republic.

But this doesn't bother the Leftists of the Democratic Party at all. They are too busy counting the loot. And there is loot to be had.

The American Left receives many things from their conquest, not the least of which is the damage it does, axiomatically, to the United States. They also receive money, lots of it especially when you think of the low funding levels the American Left has had for most of its existence. Their plunder also includes electoral organizations -- many -- as well as access to local, state, and national unions in the public and private realms. Add in mailing lists tens of millions of names long as well as websites and online acolytes by the thousands.

Most important of all, the American Left now has open access and control over sitting Democrats in Washington and the state legislatures. With money and organizations to win elections, the American Left now has the power over elected Democrats to instruct them to support and advance some decidedly non-centrist, non-liberal, but classic Leftist agendas. In a very real sense, the conquest of the Democratic Party gives the American Left a base that it could never hope to win, and will now probably never win, at the ballot box.

If you would see the American Left's apotheosis, regard the near vertical ascent, wreathed in light, of Barack H. Obama to hover over hordes of rapt worshipers eager to have their sins washed clean by his voice and visage.

Even though this regrettable transformation of the Democratic Party leaves it much smaller than it would otherwise be, it makes the American Left much bigger than it ever thought it could be. Those who have lingered all these years in the thick bong smoke of the 60s now have their fantasy within their grasp. They have made the enemies of George Bush and the New America at home and abroad into their friends and it is, at last, "Springtime for Lefties!"

Of course, it is a crowning irony to note that the proverb, "The enemy of my enemy is my friend." was originally an Arab proverb, as were, indeed, the fuming chillums of 1968's Not-So-Great-Generation.

But hey, as me and my hardcore leftist friends said way back then, "Smoke 'em if you got 'em.".
First published 2006-09-08

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