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7.3: Applying the Laws of Exponents to Rational Exponents

Difficulty Level: At Grade Created by: CK-12
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Practice Operations with Rational Exponents
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The period (in seconds) of a pendulum with a length of L (in meters) is given by the formula P=2π(L9.8)12. If the length of a pendulum is 9.883, what is its period?


When simplifying expressions with rational exponents, all the laws of exponents that were learned in the Polynomial Functions chapter are still valid. On top of that, the rules of fractions apply as well.

Example A

Simplify x12x34.

Solution: Recall from the Product Property of Exponents, that when two numbers with the same base are multiplied we add the exponents. Here, the exponents do not have the same base, so we need to find a common denominator and then add the numerators.


This rational exponent does not reduce, so we are done.

Example B

Simplify 4x23y416x3y56

Solution: This problem utilizes the Quotient Property of Exponents. Subtract the exponents with the same base and reduce 416.


If you are writing your answer in terms of positive exponents, your answer would be y1964x73. Notice, that when a rational exponent is improper we do not change it to a mixed number.

If we were to write the answer using roots, then we would take out the whole numbers. For example, y=196 can be written as y196=y3y16=y3y6 because 6 goes into 19, 3 times with a remainder of 1.

Example C

Simplify (2x12y6)234x54y94.

Solution: On the numerator, the entire expression is raised to the 23 power. Distribute this power to everything inside the parenthesis. Then, use the Powers Property of Exponents and rewrite 4 as 22.


Combine like terms by subtracting the exponents.


Finally, rewrite the answer with positive exponents by moving the 2 and x into the denominator. y74243x1112

Intro Problem Revisit Substitute 9.883 for L and solve.


Therefore, the period of the pendulum is P=2π(9.8)56.

Guided Practice

Simplify each expression. Reduce all rational exponents and write final answers using positive exponents.

1. 4d35813d25

2. w74w12

3. (332x4y65)43


1. Change 4 and 8 so that they are powers of 2 and then add exponents with the same base.


2. Subtract the exponents. Change the 12 power to 24.


3. Distribute the 43 power to everything inside the parenthesis and reduce.


Explore More

Simplify each expression. Reduce all rational exponents and write final answer using positive exponents.

  1. 15a452532a35
  2. 7b434912b23
  3. m89m23
  4. x47y116x114y53
  5. 853r5s34t1324r215s2t79
  6. (a32b45)103
  7. (5x57y4)32
  8. 4x259y4552
  9. 75d1853d3552
  10. 8132a38a9213
  11. 2723m45n32412m23n85
  12. 3x38y255x14y3102
  13. Rewrite your answer from Problem #1 using radicals.
  14. Rewrite your answer from Problem #4 using radicals.
  15. Rewrite your answer from Problem #4 using one radical.

Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 7.3. 

Image Attributions


Difficulty Level:

At Grade


Date Created:

Mar 12, 2013

Last Modified:

Jun 04, 2015
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