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2.2: Rational Exponents

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Using Rational Exponents

Radical expressions can also be written without using the radical symbol. We can use rational (fractional) exponents. The index must be a positive integer. If the index

is even, then a cannot be negative.

a1n=na

We can also have rational exponents with numerators other than In these cases, the exponent must be a fraction in lowest terms. We raise the base to a power and take an nth root. The numerator tells us the power and the denominator tells us the root.

All of the properties of exponents that we learned for integer exponents also hold for rational exponents.

Note: Rational Exponents

Rational exponents are another way to express principal nth roots. The general form for converting between a radical expression with a radical symbol and one with a rational exponent is

amn=(na)m=nam

How to: Given an expression with a rational exponent, write the expression as a radical
  1. Determine the power by looking at the numerator of the exponent.
  2. Determine the root by looking at the denominator of the exponent.
  3. Using the base as the radicand, raise the radicand to the power and use the root as the index.
Example : Writing Rational Exponents as Radicals

Write 34323 as a radical. Simplify.

Solution

The 2 tells us the power and the 3 tells us the root.

34323=(3343)2=33432

We know that 3343=7 because 73=343. Because the cube root is easy to find, it is easiest to find the cube root before squaring for this problem. In general, it is easier to find the root first and then raise it to a power.

34323=(3343)2=72=49

 

Example Writing Rational Exponents as Radicals

Write 47a2 using a rational exponent.

Solution

The power is 2 and the root is 7, so the rational exponent will be 27. We get 4a27. Using properties of exponents, we get 472=4a27

Example : Simplifying Rational Exponents

Simplify:

a. 5(2x34)(3x15)

b. (169)12

Solution

a.

30x34x15    Multiply the coefficients

30x34+15    Use properties of exponents

30x1920    Simplify

(916)12 Use definition of negative exponents

916 Rewrite as a radical

916 Use the quotient rule

34 Simplify


2.2: Rational Exponents is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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