8.1 Rational Exponents And Radicals

Unit: Roots and Radicals

Module: Rational Exponents and Radicals

Radical Notation and Properties of Roots

• • • •

n

x = x1 n when x ≥ 0, n > 0

n

a n = a if n is odd and greater than zero

n

a n is always positive if n is even and greater than zero.

A0 = 1 if A ≠ 0

You will often find it useful to express roots as exponents. In doing this, it turns out that the root becomes a fraction with the root index as the denominator. This example shows how the square root of 16 equals 16 to the ½ power. In this example the root 3 becomes the denominator in a rational exponent. Note how the exponent 15 becomes the numerator. Then the fraction 15/3 is reduced, because that is possible in this case. The simplified result shows that the original 5 statement equals 7 . Work with negative bases very carefully. Taking the odd root of a negative number produces a negative value. Taking the even root of a negative number is a different prospect. • There is no even root for a negative number or a negative number taken to an odd power. • The root of a negative number taken to an even power will be a positive number. One specific exponent must be noted, 0. Any number or variable to the 0 power equals 1. The only way to get an exponent 0 occurs when an exponent subtracts from itself. That happens in division and only when a quantity divides with itself. Any number divided by itself equals 1. NOTE: A cannot equal 0 or the division cannot occur.

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Unit: Roots and Radicals

Module: Rational Exponents and Radicals

Variables and Negative Values Under a Radical • •

Even roots of negative numbers do not exist in the real number system. Roots can be expressed as fractional exponents.

One area of special note about roots involves the roots of negative numbers. No even root exists for a negative number. There is no way to multiply any number with itself an even number of times and produce a negative number. The converse is that no negative number will have a real number even root. Odd roots for negative numbers are not a problem. Any negative number can multiply with itself an odd number of times and produce a negative number. The converse is that any negative number will have a real number odd root. Finding the roots of variables is a matter of dividing the root into the variables’ exponents. This example shows clearly that either the radical or the rational form of the root can be used to derive the root. Notice again that the root is the denominator of the rational exponent.

Here is another example showing that taking the cube root of a number is a matter of dividing the exponent of the number by 3. 3

125 turns out to equal 5 . When that exponent is 1 divided by 3, the result is 5 or 5, as shown.

This final example involves a negative coefficient. First notice that the requested root is an odd number so a negative root can be expected. 5

The fifth root of –1 is –1, that of m is m, and that of 20/5 4 n is n .

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Unit: Roots and Radicals

Module: Rational Exponents and Radicals

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Converting Rational Exponents and Radicals

• •

The radical is the symbol ( ) indicating you are to find a root. The desired root is indicated by the number written in the “v” of the sign. A radical with no number in the “v” is asking for a square, or second, root. m n

Rational Exponents are exponents in fraction form, a . The fraction exponent is m

the equivalent of the radical notation, a n = ( n a ) m . The denominator states the root of the base. The numerator indicates the power of the base.

•

Taking roots of numbers: To find the root of a number, determine the number that originally multiplied with itself some number of times to give you the number being 1 3

“rooted.” For example: 8 is asking: 1

x . x . x = 8, what is x? The answer is 2, so the cube root of 8 is 2, or 8 3 = 2.

The exponent ½ asks for the square root of a number. It wants to know what number multiplied with itself will produce the base number. The exponent 1/3 asks for the cube root of a number. It wants to know what number multiplied with itself and then multiplied again will produce the base number. NOTE: The square root and the rational exponent ½ are asking for the same thing.

In algebra, it’s generally easier to work with a problem if you rewrite the radicals using rational exponents.

REMEMBER: There is no negative number that has a square root. This is true because there is no number that when you multiply it with itself gets a negative answer.

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