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College Preparatory Mathematics

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Section 8.6 Rational Exponents

Objective: Convert between radical notation and exponential notation and simplify expressions with rational exponents using the properties of exponents.
When we simplify radicals with exponents, we divide the exponent by the index. Another way to write division is with a fraction bar. This idea is how we will define rational exponents.
Definition of Rational Exponents: anm=(am)n
The denominator of a rational exponent becomes the index on our radical, likewise the index on the radical becomes the denominator of the exponent. We can use this property to change any radical expression into an exponential expression.

Example 8.6.1.

(x5)3=x35 (3x6)5=(3x)56 Index is denominator
1(a7)3=a37 1(xy3)2=(xy)23 Negative exponents from reciprocals
We can also change any rational exponent into a radical expression by using the denominator as the index.

Example 8.6.2.

a53=(a3)5 (2mn)27=(2mn7)2 Index is denominator
x45=1(x5)4 (xy)29=1(xy9)2 Negative exponents means reciprocals
World View Note: Nicole Oresme, a Mathematician born in Normandy was the first to use rational exponents. He used the notation 139p to represent 913. However his notation went largely unnoticed.
The ability to change between exponential expressions and radical expressions allows us to evaluate problems we had no means of evaluating before by changing to a radical.

Example 8.6.3.

2743Change to radical, denominator is index, negative means reciprocal1(273)4Evaluate radical1(3)4Evaluate exponent181Our Solution
The largest advantage of being able to change a radical expression into an exponential expression is we are now allowed to use all our exponent properties to simplify. The following table reviews all of our exponent properties.
Properties of Exponents 
aman=am+n(ab)m=ambmam=1amaman=amn(ab)m=ambm1am=am(am)n=amna0=1(ab)m=bmam
When adding and subtracting with fractions we need to be sure to have a common denominator. When multiplying we only need to multiply the numerators together and denominators together. The following examples show several different problems, using different properties to simplify the rational exponents.

Example 8.6.4.

a23b12a16b15Need common denominator on a’s (6) and b’s (10)a46b510a16b210Add exponents on a’s and b’sa56b710Our Solution

Example 8.6.5.

(x13y25)34Multiply 34 by each exponentx14y310Our Solution

Example 8.6.6.

x2y232x12y56x72y0In numerator, need common denominator to add exponentsx42y462x12y56x72y0Add exponents in numerator, in denominator, y0=12x52y96x72Subtract exponents on x, reduce exponent on y2x1y32Negative exponent moves down to denominator2y32xOur Solution

Example 8.6.7.

(25x13y259x45y32)12Using order of operations, simplify inside parenthesis first. Need common denominators before we can subtract exponents(25x515y4109x1215y1510)12Subtract exponents, be careful of the negative: 410(1510)=410+1510=1910(25x715y19109)12The negative exponent will flip the fraction(925x715y1910)12The exponent 12 goes on each factor9122512x730y1920Evaluate 912 and 2512 and move negative exponent3x7305y1920Our Solution
It is important to remember that as we simplify with rational exponents we are using the exact same properties we used when simplifying integer exponents. The only difference is we need to follow our rules for fractions as well. It may be worth reviewing your notes on exponent properties to be sure your comfortable with using the properties.

Exercises Exercises – Rational Exponents

Exercise Group.

Write each expression in radical form.

Exercise Group.

Write each expression in exponential form.

Exercise Group.

Evaluate.

Exercise Group.

Simplify. Your answer should contain only positive exponents.
24.
2x2y53x54y53xy12
31.
(x43y13y1)x13y2
32.
(x12y0)43y4x2y23