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In this lesson, it will be proven that the properties of exponents are also valid for rational exponents. Moreover, these properties will be used to simplify numeric and algebraic expressions with rational exponents and roots.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
- Product of Powers Property
- Quotient of Powers Property
- Power of a Power Property
- Power of a Product Property
- Power of a Quotient Property
- Rational Exponents
Here are a few practice exercises before getting started with this lesson.
a Use the properties of exponents for integer numbers to match each algebraic expression on the left with its equivalent expression on the right.
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b Write using a rational exponent.
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c Write as a radical expression.
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Challenge
Rewriting an Algebraic Expression
Consider the following algebraic expression.
Can the above expression be written using only rational exponents? Using only radicals?
Discussion
Properties of Exponents
The following table shows the Properties of Exponents.
| Product of Powers Property | |
|---|---|
| Quotient of Powers Property | |
| Power of a Product Property | |
| Power of a Quotient Property | |
| Power of a Power Property |
Discussion
Product of Powers Property for Rational Exponents
The multiplication of two powers with the same base and rational exponents and results in a power with base and exponent
The expression can be undefined for some non-positive values of Therefore, this rule will only be defined for positive values of
Proof
Product of Powers Property for Rational ExponentsSince and are rational numbers, they can be written as the quotients of two integers. Let and be three integers such that is the quotient of and and is the quotient of and
Using the above definitions, it can be proven that is equal to
Recall that and are integer numbers. Therefore, the Product of Powers Property can be used.
The resulting expression can be written using only exponents.
It has been proven that for rational numbers and
Discussion
Quotient of Powers Property for Rational Exponents
The quotient of two powers with same base and rational exponents and results in a power with base and exponent
The expression can be undefined for some non-positive values of Therefore, this rule will only be defined for positive values of
Proof
Quotient of Powers Property for Rational ExponentsSince and are rational numbers, they can be written as the quotients of two integers. Let and be three integers such that is the quotient of and and is the quotient of and
Using the above equalities, can be proven to be equal to
Recall that and are integer numbers. Therefore, the Quotient of Powers Property can be used.
The resulting expression can be written using only exponents.
It has been proven that for rational numbers and
Discussion
Power of a Product Property for Rational Exponents
Multiplying two numbers or variables and then raising the product to the power of where is rational, is the same as raising the factors to the power of and then multiplying them.
The expression can be undefined for some non-positive values of and Therefore, this rule will only be defined for positive values of and
Proof
Power of a Product Property for Rational ExponentsSince is a rational number, it can be written as the quotient of two integers and where
Using the above equality, can be proven to be equal to
Recall that is an integer number. Therefore, the Power of a Product Property can be used to rewrite the expression.
The index of the above radical is an integer number, Therefore, the obtained expression can be rewritten as the product of two radicals. Then, the definition of a rational exponent can be used.
It has been proven that for rational
Discussion
Power of a Quotient Property for Rational Exponents
Dividing two numbers or variables and then raising the quotient to the power of where is rational, is the same as raising the divisor and dividend to the power of and then calculating the quotient.
The expression can be undefined for some non-positive values of and Therefore, this rule will only be defined for positive values of and
Proof
Power of a Quotient Property for Rational ExponentsSince is a rational number, it can be written as the quotient of two integers and where
Using the above equality, it can be proven that is equal to
Recall that is an integer number. Therefore, the Power of a Quotient Property can be used to rewrite the expression.
The index of the above radical is an integer number. Therefore, the obtained expression can be rewritten as the quotient of two radicals. Then, the definition of a rational exponent can be used.
It has been proven that for rational
Discussion
Power of a Power Property for Rational Exponents
If a power with base and rational exponent is raised to the power of where is rational, then the result is a power with base and exponent
The expression can be undefined for some non-positive values of Therefore, this rule will only be defined for positive values of
Proof
Power of a Power Property for Rational ExponentsSince and are rational numbers, they can be written as the quotients of two integers. Let and be three integers such that is the quotient of and and is the quotient of and
Using the above definitions, it can be proven that is equal to
Recall that can be defined either as or Using the second definition, the last obtained expression can be rewritten.
The root of a root can be expressed using only one root whose index is the product of the original indices.
Note that since is an integer number, then is also integer. Using the fact that and represent the same expression the power can be moved inside the radical.
As and are integer numbers, the Power of a Power Property can be used.
It has been proven that for rational numbers and
Example
Simplifying a Numeric Expression
Classmates Vincenzo and Magdalena have each simplified the same numeric expression Yet, they obtained different results. 
Use the Product of Powers Property for Rational Exponents to determine who is correct.
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Solution
The exponent of the product is the sum of the exponents of the factors. In the given expression, the exponents are fractions. To add two fractions and determine the exponent of the product, find equivalent fractions with the same denominator.
Magdalena's answer is correct. To see if Vincenzo's answer is correct, rewrite the obtained expression as a radical.
Vincenzo's answer is also correct! Therefore, both Magdalena and Vincenzo are correct.
Example
Simplifying the Powers in an Algebraic Expression
Dylan was asked to simplify the expression and write the answer using rational exponents for a homework assignment. However, Dylan did not pay attention during the lesson and now he has no clue how to find the answer. Use the Product of Powers Property for Rational Exponents and find the answer to help Dylan with his homework!
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Hint
Start by using the Commutative Property of Multiplication.
Solution
First, the Commutative Property of Multiplication can be used to rearrange the expression. Then, the Product of Powers Property can be applied.
Example
Simplifying the Quotient of Two Powers
Vincenzo and Magdalena continue with their homework, and this time, they are asked to simplify the numeric expression Once again, they obtained different results!
Use the Quotient of Powers Property for Rational Exponents to determine who is correct.
{"type":"choice","form":{"alts":["Neither","Both","Vincenzo","Magdalena"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}
Solution
The exponent of the quotient is the difference between the exponents of the dividend and the divisor. In the given expression, the exponents are fractions. To subtract two fractions, find equivalent fractions with the same denominator.
Since Vincenzo's answer is he is not correct. To see if Magdalena is correct, the obtained expression needs to be rewritten as a radical.
Since Magdalena wrote she is not correct. Therefore, neither Vincenzo nor Magdalena obtained the correct answer this time.
Example
Simplifying a Rational Expression Containing Products and a Quotient
Dylan is making progress with his homework. In one of the exercises, he is asked to simplify the given expression.
Help Dylan to complete the given task and then write the answer using rational exponents.
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Hint
Start by using the Commutative Property of Multiplication. Then, use the Product of Powers Property for Rational Exponents. Finally, apply the Quotient of Powers Property for Rational Exponents.
Solution
First, the Commutative Property of Multiplication can be used to rearrange the expression. Then, the Product of Powers Property can be used to simplify the numerator. Finally, the Quotient of Powers Property can be used to fully simplify the expression.
CommutativePropMult
Commutative Property of Multiplication
▼
Add fractions
WriteProdFrac
Write as a product of fractions
▼
Subtract fractions
Example
Matching Equivalent Rational Expressions
Dylan is trying to finish the last few exercises of his homework. Help him get a passing grade by matching the equivalent expressions!
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Hint
Use and combine the properties of exponents presented in this lesson.
Solution
The properties of rational exponents can be combined to simplify algebraic or numeric expressions. Analyze each of the given expressions one at a time.
Simplifying the First Expression
Consider the first expression.Simplifying the Second Expression
Consider now the second expression.Simplifying the Third Expression
Now, consider the third expression.Simplifying the Fourth Expression
Finally, consider the last expression.▼
Write radicals as powers
CommutativePropMult
Commutative Property of Multiplication
AddFrac
Add fractions
WriteProdFrac
Write as a product of fractions
▼
Simplify
Pop Quiz
Practicing the Power of a Power Property for Rational Expressions
Use the Power of a Power Property for Rational Exponents to calculate the exponent after simplifying the given expression. Please enter only the exponent into the answer box and write it as a fraction. Even if it is possible, do not simplify the fraction.
Note that the properties of rational exponents are helpful for almost every field in math.
Example
Using Properties of Rational Exponents to Rewrite Formulas
Below are the formulas for the surface area and the volume of a sphere with radius
a Express the radius in terms of the surface area Write the answer using rational exponents in the simplest form.
b Using the answer for Part A, express the volume in terms of the surface area Write the answer using rational exponents in the simplest form.
Answer
a
b
Hint
a Use inverse operations to isolate in the surface area formula.
b In the volume formula, substitute the expression for obtained in Part A. Then, simplify as much as possible.
Solution
a Use inverse operations to isolate on one side of the equation.
b In the formula for volume, substitute the expression for obtained in Part A and simplify as much as possible.
CalcPow
Calculate power
▼
Multiply
WriteProdFrac
Write as a product of fractions
MultFrac
Multiply fractions
Closure
Applying Properties of Exponents to Simplify an Expression
In this lesson, the understanding of the properties of exponents was extended to include rational exponents. Using these properties, the challenge presented at the beginning of the lesson can now be solved.
Simplify the expression and write the answer using only rational exponents.
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Solution
Start by rewriting the radicals as powers with rational exponents.
Now, the properties studied in this lesson can be used to simplify the expression.
Rewrite
Rewrite as
▼
Multiply
CommutativePropMult
Commutative Property of Multiplication
WriteProdFrac
Write as a product of fractions
▼
Subtract terms
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