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Exponential Functions

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

Root

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

The resulting number is commonly called a radical. For example, the radical expression is the fourth root of simplifies to because multiplied by itself times equals The general expression represents a number which equals when multiplied by itself times.

For any real number and natural number the expression is defined as the root of Note that a root with an even index is defined only for non-negative numbers. Therefore, if is even, then must be non-negative.
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Just as with exponents, the most common roots have special names: square roots and cube roots have an index of and 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.

Notice that the denominator of the rational exponent gives the index of the root, while the numerator gives the power to which is raised. The Properties of Exponents apply to rational exponents in the same way they apply to integers. Consider
Thus, is equivalent to
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Exercise

Rewrite the given expressions in the opposite form.

Show Solution
Solution
Example

To begin, notice that the expression is written in radical form. Thus, it needs to be rewritten with a rational exponent. Recall that Notice that in does not have an exponent. That means it is raised to the power of This gives

Example

This expression has a rational exponent. Thus, we must rewrite it with a radical. Recall that Since the denominator of the exponent is we can write the cube root of

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Exercise

Simplify the following radical expression by rewriting it so that it has a rational exponent.

Show Solution
Solution
Recall the following rule. 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.
Thus, the expression equals


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