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By the end of this section, you will be able to:
  • Simplify expressions with a 1 n
  • Simplify expressions with a m n
  • Use the Laws of Exponents to simply expressions with rational exponents

Before you get started, take this readiness quiz.

  1. Add: 7 15 + 5 12 .
    If you missed this problem, review [link] .
  2. Simplify: ( 4 x 2 y 5 ) 3 .
    If you missed this problem, review [link] .
  3. Simplify: 5 −3 .
    If you missed this problem, review [link] .

Simplify expressions with a 1 n

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.

The Power Property for Exponents says that ( a m ) n = a m · n when m and n are whole numbers. Let’s assume we are now not limited to whole numbers.

Suppose we want to find a number p such that ( 8 p ) 3 = 8 . We will use the Power Property of Exponents to find the value of p .

( 8 p ) 3 = 8 Multiply the exponents on the left. 8 3 p = 8 Write the exponent 1 on the right. 8 3 p = 8 1 The exponents must be equal. 3 p = 1 Solve for p . p = 1 3 So ( 8 1 3 ) 3 = 8 .

But we know also ( 8 3 ) 3 = 8 . Then it must be that 8 1 3 = 8 3 .

This same logic can be used for any positive integer exponent n to show that a 1 n = a n .

Rational exponent a 1 n

If a n is a real number and n 2 , a 1 n = a n .

There will be times when working with expressions will be easier if you use rational exponents    and times when it will be easier if you use radicals. In the first few examples, you’ll practice converting expressions between these two notations.

Write as a radical expression: x 1 2 y 1 3 z 1 4 .

Solution

We want to write each expression in the form a n .


x 1 2 The denominator of the exponent is 2, so the index of the radical is 2. We do not show the index when it is 2. x


y 1 3 The denominator of the exponent is 3, so the index is 3. y 3


z 1 4 The denominator of the exponent is 4, so the index is 4. z 4

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Write as a radical expression: t 1 2 m 1 3 r 1 4 .

t m 3 r 4

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Write as a radial expression: b 1 2 z 1 3 p 1 4 .

b z 3 p 4

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Write with a rational exponent: x y 3 z 4 .

Solution

We want to write each radical in the form a 1 n .


x No index is shown, so it is 2. The denominator of the exponent will be 2. x 1 2


y 3 The index is 3, so the denominator of the exponent is 3. y 1 3


z 4 The index is 4, so the denominator of the exponent is 4. z 1 4

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Write with a rational exponent: s x 3 b 4 .

s 1 2 x 1 3 b 1 4

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Write with a rational exponent: v p 3 p 4 .

v 1 2 p 1 3 p 1 4

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Write with a rational exponent: 5 y 4 x 3 3 5 z 4 .

Solution

We want to write each radical in the form a 1 n .


  1. 5 y No index is shown, so it is 2. The denominator of the exponent will be 2. ( 5 y ) 1 2


  2. 4 x 3 The index is 3, so the denominator of the exponent is 3. ( 4 x ) 1 3


  3. 3 5 z 4 The index is 4, so the denominator of the exponent is 4. 3 ( 5 z ) 1 4
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Write with a rational exponent: 10 m 3 n 5 3 6 y 4 .

( 10 m ) 1 2 ( 3 n ) 1 5 ( 486 y ) 1 4

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Write with a rational exponent: 3 k 7 5 j 4 8 2 a 3 .

( 3 k ) 1 7 ( 5 j ) 1 4 ( 1024 a ) 1 3

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In the next example, you may find it easier to simplify the expressions if you rewrite them as radicals first.

Simplify: 25 1 2 64 1 3 256 1 4 .

Solution


25 1 2 Rewrite as a square root. 25 Simplify. 5


64 1 3 Rewrite as a cube root. 64 3 Recognize 64 is a perfect cube. 4 3 3 Simplify. 4


256 1 4 Rewrite as a fourth root. 256 4 Recognize 256 is a perfect fourth power. 4 4 4 Simplify. 4

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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