Rewriting Expressions with Radicals and Rational Exponents

Radical expressions involve taking the root of a quantity. These expressions can be expressed with fraction exponents, or rational exponents. Since radicals and rational exponents are two different ways to write the same thing, translating between the two is useful.

Concept

$n^{th}$ Root

The $n^{th}$ root of a number $a$ expresses another number that, when multiplied by itself $n$ times, will result in $a .$ Aside from the radical symbol, the notation is made up of the radicand $a$ and the index $n .$

The resulting number is commonly called a radical. For example, the radical expression $416$ is the fourth root of $16 .$ $416$ simplifies to $2$ because $2$ multiplied by itself $4$ times equals $16 .$ $416 = 4 2^{4} = 2$ The general expression $n a$ represents a number which equals $a$ when multiplied by itself $n$ times.

$n times n a \cdot n a \cdot \dots \cdot n a = a or (n a)^{n} = a$

For any real number

a

and natural number

n,

the expression

a^{\frac{1}{n}}

is defined as the

n^{th}

root of

a .

Note that a root with an even index is defined only for non-negative numbers. Therefore, if

n

is even, then

a

must be non-negative.

Just as with exponents, the most common roots have special names: square roots and cube roots have an index of

2

and

3,

respectively.

Rule

Rational Exponents

When a number is raised to the power of a fraction, that fraction is the number's rational exponent. Such an expression is equivalent to a root.

$a^{\frac{m}{n}} = n a^{m}$

Notice that the denominator of the rational exponent gives the index of the root, while the numerator gives the power to which

a

is raised. The Properties of Exponents apply to rational exponents in the same way they apply to integers. Consider

5 8^{2} .

5 8^{2}

RootToPowD

$n a = a^{\frac{1}{n}}$

(8^{2})^{\frac{1}{5}}

PowPow

$(a^{m})^{n} = a^{m \cdot n}$

8^{2 \cdot \frac{1}{5}}

MoveLeftFacToNumOne

$a \cdot \frac{1}{b} = \frac{a}{b}$

8^{\frac{2}{5}}

Thus,

5 8^{2}

is equivalent to

8^{\frac{2}{5}} .

Rewrite between rational exponent form and radical form

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Exercise

Rewrite the given expressions in the opposite form. $5 x and x^{\frac{2}{3}}$

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Solution

Example

$5 x$

To begin, notice that the expression

5 x

is written in radical form. Thus, it needs to be rewritten with a rational exponent. Recall that

n a^{m} = a^{\frac{m}{n}} .

Notice that in

5 x,

x

does not have an exponent. That means it is raised to the power of

1 .

This gives

5 x = x^{\frac{1}{5}} .

Example

$x^{\frac{2}{3}}$

This expression has a rational exponent. Thus, we must rewrite it with a radical. Recall that

a^{\frac{m}{n}} = n a^{m} .

Since the denominator of the exponent is

3,

we can write the cube root of

x^{2} .

x^{\frac{2}{3}} = 3 x^{2}

Simplify the radical expression

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Exercise

Simplify the following radical expression by rewriting it so that it has a rational exponent. $11 5^{33}$

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Solution

Recall the following rule.

n a^{m} = a^{\frac{m}{n}}

Using this, it is possible to rewrite a radical expression so that it has a rational exponent. We will first use this rule and then simplify the resulting expression.

11 5^{33}

$n a^{m} = a^{\frac{m}{n}}$

5^{\frac{3 3}{1 1}}

SimpQuot

Simplify quotient

5^{3}

CalcPow

Calculate power

125

Thus, the expression equals

125 .