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Algebra 1 View details

2. Extending the Properties of Exponents to Rational Exponents

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Chapter 6

2.

Extending the Properties of Exponents to Rational Exponents

Rational exponents provide a bridge between roots and powers in mathematics. By understanding the properties of rational exponents, learners can handle complex expressions with ease. In practical scenarios, these properties offer shortcuts for simplifying calculations and streamlining problem-solving. Engineers, scientists, and researchers often leverage these properties to optimize algorithms, making computational tasks more efficient. Grasping these properties is thus a stepping stone to deeper insights and enhanced mathematical proficiency.

Lesson Settings & Tools

	16 Theory slides
	10 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

Extending the Properties of Exponents to Rational Exponents

Slide of 16

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.

b Write

4 2^{3}

using a rational exponent.

c Write

3^{\frac{4}{5}}

as a radical expression.

Challenge

Rewriting an Algebraic Expression

Consider the following algebraic expression.

\frac{8 x ^{3} \cdot ( x ^{\frac{1}{2}} y ) ^{5}}{3 2 x ^{2} y \cdot x ^{\frac{1}{3}}}

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	$a^{m} a^{n} = a^{m + n}$
Quotient of Powers Property	$\frac{a ^{m}}{a ^{n}} = a^{m - n}$
Power of a Product Property	$(a b)^{n} = a^{n} b^{n}$
Power of a Quotient Property	$(\frac{a}{b})^{n} = \frac{a ^{n}}{b ^{n}}$
Power of a Power Property	$(a^{m})^{n} = a^{m \cdot n}$

These properties hold true for integer exponents. In this lesson, they will be shown to be true for rational exponents as well.

Discussion

Product of Powers Property for Rational Exponents

The multiplication of two powers with the same base

a

and rational exponents

m

and

n

results in a power with base

a

and exponent

m + n .

sum of exponents

The expression can be undefined for some non-positive values of

a .

Therefore, this rule will only be defined for positive values of

a .

Proof

Product of Powers Property for Rational Exponents

Since

m

and

n

are rational numbers, they can be written as the quotients of two integers. Let

p,

q,

and

r

be three integers such that

r \neq = 0,

m

is the quotient of

p

and

r,

and

n

is the quotient of

q

and

r .

m = \frac{p}{r} and n = \frac{q}{r}

Using the above definitions, it can be proven that

a^{m} \cdot a^{n}

is equal to

a^{m + n} .

a^{m} \cdot a^{n}

$m = \frac{p}{r}$ , $n = \frac{q}{r}$

a^{\frac{p}{r}} \cdot a^{\frac{q}{r}}

▼

Simplify

$a^{\frac{m}{n}} = n a^{m}$

r a^{p} \cdot r a^{q}

$r a \cdot r b = r a \cdot b$

r a^{p} \cdot a^{q}

Recall that

p

and

q

are integer numbers. Therefore, the Product of Powers Property can be used.

r a^{p} \cdot a^{q} = r a^{p + q}

The resulting expression can be written using only exponents.

r a^{p + q}

▼

Simplify

$n a^{m} = a^{\frac{m}{n}}$

a^{\frac{p + q}{r}}

Write as a sum of fractions

a^{\frac{p}{r} + \frac{q}{r}}

$\frac{p}{r} = m$ , $\frac{q}{r} = n$

a^{m + n}

It has been proven that

a^{m} \cdot a^{n} = a^{m + n}

for rational numbers

m

and

n .

Discussion

Quotient of Powers Property for Rational Exponents

The quotient of two powers with same base

a

and rational exponents

m

and

n

results in a power with base

a

m - n .

difference of exponents

The expression can be undefined for some non-positive values of

a .

Therefore, this rule will only be defined for positive values of

a .

Proof

Quotient of Powers Property for Rational Exponents

Since

m

and

n

are rational numbers, they can be written as the quotients of two integers. Let

p,

q,

and

r

be three integers such that

r \neq = 0,

m

is the quotient of

p

and

r,

and

n

is the quotient of

q

and

r .

m = \frac{p}{r} and n = \frac{q}{r}

Using the above equalities,

\frac{a ^{m}}{a ^{n}}

can be proven to be equal to

a^{m - n} .

\frac{a ^{m}}{a ^{n}}

$m = \frac{p}{r}$ , $n = \frac{q}{r}$

\frac{a ^{\frac{p}{r}}}{a ^{\frac{q}{r}}}

▼

Simplify

$a^{\frac{m}{n}} = n a^{m}$

\frac{r a ^{p}}{r a ^{q}}

$\frac{r a}{r b} = r \frac{a}{b}$

r \frac{a ^{p}}{a ^{q}}

Recall that

p

and

q

are integer numbers. Therefore, the Quotient of Powers Property can be used.

r \frac{a ^{p}}{a ^{q}} = r a^{p - q}

The resulting expression can be written using only exponents.

r a^{p - q}

▼

Simplify

$n a^{m} = a^{\frac{m}{n}}$

a^{\frac{p - q}{r}}

Write as a difference of fractions

a^{\frac{p}{r} - \frac{q}{r}}

$\frac{p}{r} = m$ , $\frac{q}{r} = n$

a^{m - n}

It has been proven that

\frac{a ^{m}}{a ^{n}} = a^{m - n}

for rational numbers

m

and

n .

Be aware that

a \div b

is the same as

\frac{a}{b} .

Therefore,

a^{m} \div a^{n}

is also equal to

a^{m - n},

when

a > 0

and both

m

and

n

are rational numbers.

Discussion

Power of a Product Property for Rational Exponents

Multiplying two numbers or variables and then raising the product to the power of

n,

where

n

is rational, is the same as raising the factors to the power of

n

and then multiplying them.

Power of a product

The expression can be undefined for some non-positive values of

a

and

b .

Therefore, this rule will only be defined for positive values of

a

and

b .

Proof

Power of a Product Property for Rational Exponents

Since

n

is a rational number, it can be written as the quotient of two integers

p

and

q,

where

q \neq = 0 .

n = \frac{p}{q}

Using the above equality,

(a b)^{n}

can be proven to be equal to

a^{n} b^{n} .

(a b)^{n}

$n = \frac{p}{q}$

(a b)^{\frac{p}{q}}

$a^{\frac{m}{n}} = n a^{m}$

q (a b)^{p}

Recall that

p

is an integer number. Therefore, the Power of a Product Property can be used to rewrite the expression.

q (a b)^{p} = q a^{p} b^{p}

The index of the above radical is an integer number,

q .

Therefore, the obtained expression can be rewritten as the product of two radicals. Then, the definition of a rational exponent can be used.

q a^{p} b^{p}

$q a \cdot b = q a \cdot q b$

q a^{p} q b^{p}

$n a^{m} = a^{\frac{m}{n}}$

a^{\frac{p}{q}} b^{\frac{p}{q}}

$\frac{p}{q} = n$

a^{n} b^{n}

It has been proven that

(a b)^{n} = a^{n} b^{n}

for rational

n .

Discussion

Power of a Quotient Property for Rational Exponents

Dividing two numbers or variables and then raising the quotient to the power of

n,

where

n

is rational, is the same as raising the divisor and dividend to the power of

n

and then calculating the quotient.

power of a quotient

The expression can be undefined for some non-positive values of

a

and

b .

Therefore, this rule will only be defined for positive values of

a

and

b .

Proof

Power of a Quotient Property for Rational Exponents

Since

n

is a rational number, it can be written as the quotient of two integers

p

and

q,

where

q \neq = 0 .

n = \frac{p}{q}

Using the above equality, it can be proven that

(\frac{a}{b})^{n}

is equal to

\frac{a ^{n}}{b ^{n}} .

(\frac{a}{b})^{n}

$n = \frac{p}{q}$

(\frac{a}{b})^{\frac{p}{q}}

$a^{\frac{m}{n}} = n a^{m}$

q (\frac{a}{b})^{p}

Recall that

p

is an integer number. Therefore, the Power of a Quotient Property can be used to rewrite the expression.

q (\frac{a}{b})^{p} = q \frac{a ^{p}}{b ^{p}}

The index

q

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.

q \frac{a ^{p}}{b ^{p}}

$q \frac{a}{b} = \frac{q a}{q b}$

\frac{q a ^{p}}{q b ^{p}}

$n a^{m} = a^{\frac{m}{n}}$

\frac{a ^{\frac{p}{q}}}{b ^{\frac{p}{q}}}

$\frac{p}{q} = n$

\frac{a ^{n}}{b ^{n}}

It has been proven that

(\frac{a}{b})^{n} = \frac{a ^{n}}{b ^{n}}

for rational

n .

Discussion

Power of a Power Property for Rational Exponents

If a power with base

a

and rational exponent

m

is raised to the power of

n,

where

n

is rational, then the result is a power with base

a

and exponent

m \cdot n .

Multiplication of exponents

The expression can be undefined for some non-positive values of

a .

Therefore, this rule will only be defined for positive values of

a .

Proof

Power of a Power Property for Rational Exponents

Since

m

and

n

are rational numbers, they can be written as the quotients of two integers. Let

p,

q,

and

r

be three integers such that

r \neq = 0,

m

is the quotient of

p

and

r,

and

n

is the quotient of

q

and

r .

m = \frac{p}{r} and n = \frac{q}{r}

Using the above definitions, it can be proven that

(a^{m})^{n}

is equal to

a^{m \cdot n} .

(a^{m})^{n}

$m = \frac{p}{r}$ , $n = \frac{q}{r}$

(a^{\frac{p}{r}})^{\frac{q}{r}}

$a^{\frac{m}{n}} = n a^{m}$

(r a^{p})^{\frac{q}{r}}

Recall that

a^{\frac{m}{n}}

can be defined either as

n a^{m}

or

(n a)^{m} .

Using the second definition, the last obtained expression can be rewritten.

(r a^{p})^{\frac{q}{r}} = (r r a^{p})^{q}

The root of a root can be expressed using only one root whose index is the product of the original indices.

(r r a^{p})^{q} \Leftrightarrow (r \cdot r a^{p})^{q}

Note that since

r

is an integer number, then

r \cdot r

is also integer. Using the fact that

n a^{m}

and

(n a)^{m}

represent the same expression

a^{\frac{m}{n}},

the power

q

can be moved inside the radical.

(r \cdot r a^{p})^{q} \Leftrightarrow r \cdot r (a^{p})^{q}

As

p

and

q

are integer numbers, the Power of a Power Property can be used.

r \cdot r (a^{p})^{q}

$(a^{m})^{n} = a^{m \cdot n}$

r \cdot r a^{p \cdot q}

$n a^{m} = a^{\frac{m}{n}}$

a^{\frac{p \cdot q}{r \cdot r}}

Write as a product of fractions

a^{\frac{p}{r} \cdot \frac{q}{r}}

$\frac{p}{r} = m$ , $\frac{q}{r} = n$

a^{m \cdot n}

It has been proven that

(a^{m})^{n} = a^{m \cdot n}

for rational numbers

m

and

n .

Example

Simplifying a Numeric Expression

Classmates Vincenzo and Magdalena have each simplified the same numeric expression

3^{\frac{1}{4}} \cdot 3^{\frac{1}{2}} .

Yet, they obtained different results.

Vincenzo's and Magdalena's answers

Use the Product of Powers Property for Rational Exponents to determine who is correct.

Hint

The exponent of the product is the sum of the exponents of the factors.

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.

3^{\frac{1}{4}} \cdot 3^{\frac{1}{2}}

$a^{m} \cdot a^{n} = a^{m + n}$

3^{\frac{1}{4} + \frac{1}{2}}

▼

Add fractions

$\frac{a}{b} = \frac{a \cdot 2}{b \cdot 2}$

3^{\frac{1}{4} + \frac{2}{4}}

Add fractions

3^{\frac{3}{4}}

Magdalena's answer is correct. To see if Vincenzo's answer is correct, rewrite the obtained expression as a radical.

3^{\frac{3}{4}}

$a^{\frac{m}{n}} = n a^{m}$

4 3^{3}

Vincenzo's answer is also correct! Therefore, both Magdalena and Vincenzo are correct.

In the given expression, the powers were different but the bases were the same. However, what happens if different bases are involved?

Example

Simplifying the Powers in an Algebraic Expression

Dylan was asked to simplify the expression

x^{\frac{1}{2}} y^{\frac{2}{7}} \cdot x^{\frac{1}{3}} y^{\frac{3}{7}}

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!

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.

x^{\frac{1}{2}} y^{\frac{2}{7}} \cdot x^{\frac{1}{3}} y^{\frac{3}{7}}

CommutativePropMult

Commutative Property of Multiplication

x^{\frac{1}{2}} x^{\frac{1}{3}} \cdot y^{\frac{2}{7}} y^{\frac{3}{7}}

$a^{m} \cdot a^{n} = a^{m + n}$

x^{\frac{1}{2} + \frac{1}{3}} \cdot y^{\frac{2}{7} + \frac{3}{7}}

▼

Add fractions

$\frac{a}{b} = \frac{a \cdot 3}{b \cdot 3}$

x^{\frac{3}{6} + \frac{1}{3}} \cdot y^{\frac{2}{7} + \frac{3}{7}}

$\frac{a}{b} = \frac{a \cdot 2}{b \cdot 2}$

x^{\frac{3}{6} + \frac{2}{6}} \cdot y^{\frac{2}{7} + \frac{3}{7}}

Add fractions

x^{\frac{5}{6}} \cdot y^{\frac{5}{7}}

Multiply

x^{\frac{5}{6}} y^{\frac{5}{7}}

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

2^{\frac{5}{6}} \div 2^{\frac{1}{5}} .

Once again, they obtained different results!

Vincenzo and Magdalena's results

Use the Quotient of Powers Property for Rational Exponents to determine who is correct.

Hint

The exponent of the quotient equals the difference of the exponents of the dividend and the divisor.

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.

2^{\frac{5}{6}} \div 2^{\frac{1}{5}}

$a \div b = \frac{a}{b}$

\frac{2 ^{\frac{5}{6}}}{2 ^{\frac{1}{5}}}

$\frac{a ^{m}}{a ^{n}} = a^{m - n}$

2^{\frac{5}{6} - \frac{1}{5}}

▼

Subtract fractions

$\frac{a}{b} = \frac{a \cdot 5}{b \cdot 5}$

2^{\frac{2 5}{3 0} - \frac{1}{5}}

$\frac{a}{b} = \frac{a \cdot 6}{b \cdot 6}$

2^{\frac{2 5}{3 0} - \frac{6}{3 0}}

Subtract fractions

2^{\frac{1 9}{3 0}}

Since Vincenzo's answer is

2^{\frac{1}{6}},

he is not correct. To see if Magdalena is correct, the obtained expression needs to be rewritten as a radical.

2^{\frac{1 9}{3 0}}

$a^{\frac{m}{n}} = n a^{m}$

30 2^{19}

Since Magdalena wrote

6 2^{5},

she is not correct. Therefore, neither Vincenzo nor Magdalena obtained the correct answer this time.

The Product of Powers Property for Rational Exponents and the Quotient of Powers Property for Rational Exponents can be combined to simplify numeric or algebraic expressions with one or more variables.

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.

\frac{2 ^{\frac{2}{3}} x ^{\frac{1}{2}} \cdot 2 ^{\frac{3}{1 1}}}{2 ^{\frac{1 0}{1 1}} x ^{\frac{2}{9}}}

Help Dylan to complete the given task and then write the answer using rational exponents.

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.

\frac{2 ^{\frac{2}{3}} x ^{\frac{1}{2}} \cdot 2 ^{\frac{3}{1 1}}}{2 ^{\frac{1 0}{1 1}} x ^{\frac{2}{9}}}

CommutativePropMult

Commutative Property of Multiplication

\frac{2 ^{\frac{2}{3}} 2 ^{\frac{3}{1 1}} \cdot x ^{\frac{1}{2}}}{2 ^{\frac{1 0}{1 1}} \cdot x ^{\frac{2}{9}}}

$a^{m} \cdot a^{n} = a^{m + n}$

\frac{2 ^{\frac{2}{3} + \frac{3}{1 1}} \cdot x ^{\frac{1}{2}}}{2 ^{\frac{1 0}{1 1}} \cdot x ^{\frac{2}{9}}}

▼

Add fractions

$\frac{a}{b} = \frac{a \cdot 1 1}{b \cdot 1 1}$

\frac{2 ^{\frac{2 2}{3 3} + \frac{3}{1 1}} \cdot x ^{\frac{1}{2}}}{2 ^{\frac{1 0}{1 1}} \cdot x ^{\frac{2}{9}}}

$\frac{a}{b} = \frac{a \cdot 3}{b \cdot 3}$

\frac{2 ^{\frac{2 2}{3 3} + \frac{9}{3 3}} \cdot x ^{\frac{1}{2}}}{2 ^{\frac{1 0}{1 1}} \cdot x ^{\frac{2}{9}}}

Add fractions

\frac{2 ^{\frac{3 1}{3 3}} \cdot x ^{\frac{1}{2}}}{2 ^{\frac{1 0}{1 1}} \cdot x ^{\frac{2}{9}}}

Write as a product of fractions

\frac{2 ^{\frac{3 1}{3 3}}}{2 ^{\frac{1 0}{1 1}}} \cdot \frac{x ^{\frac{1}{2}}}{x ^{\frac{2}{9}}}

$\frac{a ^{m}}{a ^{n}} = a^{m - n}$

2^{\frac{3 1}{3 3} - \frac{1 0}{1 1}} \cdot x^{\frac{1}{2} - \frac{2}{9}}

▼

Subtract fractions

$\frac{a}{b} = \frac{a \cdot 3}{b \cdot 3}$

2^{\frac{3 1}{3 3} - \frac{3 0}{3 3}} \cdot x^{\frac{1}{2} - \frac{2}{9}}

$\frac{a}{b} = \frac{a \cdot 9}{b \cdot 9}$

2^{\frac{3 1}{3 3} - \frac{3 0}{3 3}} \cdot x^{\frac{9}{1 8} - \frac{2}{9}}

$\frac{a}{b} = \frac{a \cdot 2}{b \cdot 2}$

2^{\frac{3 1}{3 3} - \frac{3 0}{3 3}} \cdot x^{\frac{9}{1 8} - \frac{4}{1 8}}

Subtract fractions

2^{\frac{1}{3 3}} \cdot x^{\frac{5}{1 8}}

Multiply

2^{\frac{1}{3 3}} x^{\frac{5}{1 8}}

The properties of rational exponents can be combined to simplify more complicated expressions.

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!

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.

\frac{x ^{\frac{1}{2}} y ^{\frac{3}{4}}}{x ^{\frac{1}{3}}}

Here, the Quotient of Powers Property for Rational Exponents can be used.

\frac{x ^{\frac{1}{2}} y ^{\frac{3}{4}}}{x ^{\frac{1}{3}}}

MovePartNumRight

$\frac{a \cdot b}{c} = \frac{a}{c} \cdot b$

\frac{x ^{\frac{1}{2}}}{x ^{\frac{1}{3}}} \cdot y^{\frac{3}{4}}

$\frac{a ^{m}}{a ^{n}} = a^{m - n}$

x^{\frac{1}{2} - \frac{1}{3}} y^{\frac{3}{4}}

▼

Subtract fractions

$\frac{a}{b} = \frac{a \cdot 3}{b \cdot 3}$

x^{\frac{3}{6} - \frac{1}{3}} y^{\frac{3}{4}}

$\frac{a}{b} = \frac{a \cdot 2}{b \cdot 2}$

x^{\frac{3}{6} - \frac{2}{6}} y^{\frac{3}{4}}

Subtract fractions

x^{\frac{1}{6}} y^{\frac{3}{4}}

Simplifying the Second Expression

Consider now the second expression.

x \cdot x^{\frac{1}{5}}

Here, the radical expression can be written as a power with a rational exponent. Then, the Product of Powers Property for Rational Exponents can be applied.

x \cdot x^{\frac{1}{5}}

$a = a^{\frac{1}{2}}$

x^{\frac{1}{2}} x^{\frac{1}{5}}

$a^{m} \cdot a^{n} = a^{m + n}$

x^{\frac{1}{2} + \frac{1}{5}}

▼

Add fractions

$\frac{a}{b} = \frac{a \cdot 5}{b \cdot 5}$

x^{\frac{5}{1 0} + \frac{1}{5}}

$\frac{a}{b} = \frac{a \cdot 2}{b \cdot 2}$

x^{\frac{5}{1 0} + \frac{2}{1 0}}

Add fractions

x^{\frac{7}{1 0}}

Simplifying the Third Expression

Now, consider the third expression.

\frac{2 x \cdot 2 y}{2 ^{\frac{3}{2}}}

Here, the definition of a rational exponent will be used again to rewrite the radical expression. Then, the Product of Powers Property for Rational Exponents and the Quotient of Powers Property for Rational Exponents will be used.

\frac{2 x \cdot 2 y}{2 ^{\frac{3}{2}}}

$a = a^{\frac{1}{2}}$

\frac{2 ^{\frac{1}{2}} x \cdot 2 y}{2 ^{\frac{3}{2}}}

CommutativePropMult

Commutative Property of Multiplication

\frac{2 ^{\frac{1}{2}} 2 \cdot x y}{2 ^{\frac{3}{2}}}

$a = a^{1}$

\frac{2 ^{\frac{1}{2}} 2 ^{1} \cdot x y}{2 ^{\frac{3}{2}}}

$a^{m} \cdot a^{n} = a^{m + n}$

\frac{2 ^{\frac{1}{2} + 1} \cdot x y}{2 ^{\frac{3}{2}}}

▼

Add terms

Rewrite $1$ as $\frac{2}{2}$

\frac{2 ^{\frac{1}{2} + \frac{2}{2}} x y}{2 ^{\frac{3}{2}}}

Add fractions

\frac{2 ^{\frac{3}{2}} \cdot x y}{2 ^{\frac{3}{2}}}

MovePartNumRight

$\frac{a \cdot b}{c} = \frac{a}{c} \cdot b$

\frac{2 ^{\frac{3}{2}}}{2 ^{\frac{3}{2}}} \cdot x y

▼

Simplify

$\frac{a}{a} = 1$

1 x y

Identity Property of Multiplication

x y

Simplifying the Fourth Expression

Finally, consider the last expression.

\frac{2 x ^{\frac{1}{3}} 4 y}{2 y ^{\frac{1}{4}} 2 ^{\frac{1}{2}}}

Similarly to the previous expression, this one can be simplified by using the definition of a rational exponent and different properties of rational exponents.

\frac{2 x ^{\frac{1}{3}} 4 y}{2 y ^{\frac{1}{4}} 2 ^{\frac{1}{2}}}

▼

Write radicals as powers

$n a = a^{\frac{1}{n}}$

\frac{2 x ^{\frac{1}{3}} y ^{\frac{1}{4}}}{2 y ^{\frac{1}{4}} 2 ^{\frac{1}{2}}}

$a = a^{\frac{1}{2}}$

\frac{2 x ^{\frac{1}{3}} y ^{\frac{1}{4}}}{2 ^{\frac{1}{2}} y ^{\frac{1}{4}} 2 ^{\frac{1}{2}}}

CommutativePropMult

Commutative Property of Multiplication

\frac{2 x ^{\frac{1}{3}} \cdot y ^{\frac{1}{4}}}{2 ^{\frac{1}{2}} 2 ^{\frac{1}{2}} \cdot y ^{\frac{1}{4}}}

$a^{m} \cdot a^{n} = a^{m + n}$

\frac{2 x ^{\frac{1}{3}} \cdot y ^{\frac{1}{4}}}{2 ^{\frac{1}{2} + \frac{1}{2}} \cdot y ^{\frac{1}{4}}}

Add fractions

\frac{2 x ^{\frac{1}{3}} \cdot y ^{\frac{1}{4}}}{2 ^{1} \cdot y ^{\frac{1}{4}}}

Write as a product of fractions

\frac{2 x ^{\frac{1}{3}}}{2 ^{1}} \cdot \frac{y ^{\frac{1}{4}}}{y ^{\frac{1}{4}}}

MovePartNumRight

$\frac{a \cdot b}{c} = \frac{a}{c} \cdot b$

\frac{2}{2 ^{1}} \cdot x^{\frac{1}{3}} \cdot \frac{y ^{\frac{1}{4}}}{y ^{\frac{1}{4}}}

▼

Simplify

$a^{1} = a$

\frac{2}{2} \cdot x^{\frac{1}{3}} \cdot \frac{y ^{\frac{1}{4}}}{y ^{\frac{1}{4}}}

$\frac{a}{a} = 1$

1 \cdot x^{\frac{1}{3}} \cdot 1

Identity Property of Multiplication

x^{\frac{1}{3}}

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.

Write the exponent of the simplified expression

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 $r .$

Sphere with the formulas for surface area and volume

a Express the radius

r

in terms of the surface area

S A .

Write the answer using rational exponents in the simplest form.

b Using the answer for Part A, express the volume

V

in terms of the surface area

S A .

Write the answer using rational exponents in the simplest form.

Answer

a

r = \frac{S A ^{\frac{1}{2}}}{2 π ^{\frac{1}{2}}}

b

V = \frac{S A ^{\frac{3}{2}}}{6 π ^{\frac{1}{2}}}

Hint

a Use inverse operations to isolate

r

in the surface area formula.

b In the volume formula, substitute the expression for

r

obtained in Part A. Then, simplify as much as possible.

Solution

a Use inverse operations to isolate

r

on one side of the equation.

S A = 4 π r^{2}

▼

Solve for

r

$LHS / 4 π = RHS / 4 π$

\frac{S A}{4 π} = r^{2}

$LHS = RHS$

\frac{S A}{4 π} = r

$a = a^{\frac{1}{2}}$

(\frac{S A}{4 π})^{\frac{1}{2}} = r

$(\frac{a}{b})^{m} = \frac{a ^{m}}{b ^{m}}$

\frac{S A ^{\frac{1}{2}}}{( 4 π ) ^{\frac{1}{2}}} = r

$(a b)^{m} = a^{m} b^{m}$

\frac{S A ^{\frac{1}{2}}}{4 ^{\frac{1}{2}} π ^{\frac{1}{2}}} = r

$a^{\frac{1}{2}} = a$

\frac{S A ^{\frac{1}{2}}}{4 π ^{\frac{1}{2}}} = r

Calculate root

\frac{S A ^{\frac{1}{2}}}{2 π ^{\frac{1}{2}}} = r

Rearrange equation

r = \frac{S A ^{\frac{1}{2}}}{2 π ^{\frac{1}{2}}}

Note that when solving the equation, the principal root was taken. This is because the radius

r

must be a positive value.

b In the formula for volume, substitute the expression for

r

obtained in Part A and simplify as much as possible.

V = \frac{4}{3} π r^{3}

$r = \frac{S A ^{\frac{1}{2}}}{2 π ^{\frac{1}{2}}}$

V = \frac{4}{3} π (\frac{S A ^{\frac{1}{2}}}{2 π ^{\frac{1}{2}}})^{3}

$(\frac{a}{b})^{m} = \frac{a ^{m}}{b ^{m}}$

V = \frac{4}{3} π \frac{( S A ^{\frac{1}{2}} ) ^{3}}{( 2 π ^{\frac{1}{2}} ) ^{3}}

$(a b)^{m} = a^{m} b^{m}$

V = \frac{4}{3} π \cdot \frac{( S A ^{\frac{1}{2}} ) ^{3}}{2 ^{3} ( π ^{\frac{1}{2}} ) ^{3}}

Calculate power

V = \frac{4}{3} π \cdot \frac{( S A ^{\frac{1}{2}} ) ^{3}}{8 ( π ^{\frac{1}{2}} ) ^{3}}

$(a^{m})^{n} = a^{m \cdot n}$

V = \frac{4}{3} π (\frac{S A ^{\frac{1}{2} \cdot 3}}{8 π ^{\frac{1}{2} \cdot 3}})

MoveRightFacToNumOne

$\frac{1}{b} \cdot a = \frac{a}{b}$

V = \frac{4 π}{3} (\frac{S A ^{\frac{3}{2}}}{8 π ^{\frac{3}{2}}})

▼

Multiply

Multiply fractions

V = \frac{4 π S A ^{\frac{3}{2}}}{2 4 π ^{\frac{3}{2}}}

$\frac{a}{b} = \frac{a / 4}{b / 4}$

V = \frac{π S A ^{\frac{3}{2}}}{6 π ^{\frac{3}{2}}}

CommutativePropMult

Commutative Property of Multiplication

V = \frac{S A ^{\frac{3}{2}} π}{6 π ^{\frac{3}{2}}}

Write as a product of fractions

V = \frac{S A ^{\frac{3}{2}}}{6} (\frac{π}{π ^{\frac{3}{2}}})

$a = a^{1}$

V = \frac{S A ^{\frac{3}{2}}}{6} (\frac{π ^{1}}{π ^{\frac{3}{2}}})

$\frac{a ^{m}}{a ^{n}} = a^{m - n}$

V = \frac{S A ^{\frac{3}{2}}}{6} (π^{1 - \frac{3}{2}})

▼

Subtract terms

Rewrite $1$ as $\frac{2}{2}$

V = \frac{S A ^{\frac{3}{2}}}{6} (π^{\frac{2}{2} - \frac{3}{2}})

Subtract fractions

V = \frac{S A ^{\frac{3}{2}}}{6} (π^{- \frac{1}{2}})

NegExponentToFrac

$a^{- m} = \frac{1}{a ^{m}}$

V = \frac{S A ^{\frac{3}{2}}}{6} (\frac{1}{π ^{\frac{1}{2}}})

Multiply fractions

V = \frac{S A ^{\frac{3}{2}}}{6 π ^{\frac{1}{2}}}

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.

\frac{8 x ^{3} \cdot ( x ^{\frac{1}{2}} y ) ^{5}}{3 2 x ^{2} y \cdot x ^{\frac{1}{3}}}

Simplify the expression and write the answer using only rational exponents.

Hint

Use the definition of a rational exponent to express radicals as powers. Combine the properties seen in this lesson.

Solution

Start by rewriting the radicals as powers with rational exponents.

\frac{8 x ^{3} \cdot ( x ^{\frac{1}{2}} y ) ^{5}}{3 2 x ^{2} y \cdot x ^{\frac{1}{3}}}

$a = a^{\frac{1}{2}}$

\frac{( 8 x ^{3} ) ^{\frac{1}{2}} ( x ^{\frac{1}{2}} y ) ^{5}}{3 2 x ^{2} y \cdot x ^{\frac{1}{3}}}

$n a = a^{\frac{1}{n}}$

\frac{( 8 x ^{3} ) ^{\frac{1}{2}} ( x ^{\frac{1}{2}} y ) ^{5}}{( 2 x ^{2} y ) ^{\frac{1}{3}} x ^{\frac{1}{3}}}

Now, the properties studied in this lesson can be used to simplify the expression.

\frac{( 8 x ^{3} ) ^{\frac{1}{2}} ( x ^{\frac{1}{2}} y ) ^{5}}{( 2 x ^{2} y ) ^{\frac{1}{3}} x ^{\frac{1}{3}}}

$(a b)^{m} = a^{m} b^{m}$

\frac{8 ^{\frac{1}{2}} ( x ^{3} ) ^{\frac{1}{2}} ( x ^{\frac{1}{2}} ) ^{5} y ^{5}}{2 ^{\frac{1}{3}} ( x ^{2} ) ^{\frac{1}{3}} y ^{\frac{1}{3}} x ^{\frac{1}{3}}}

Rewrite $8$ as $2^{3}$

\frac{( 2 ^{3} ) ^{\frac{1}{2}} ( x ^{3} ) ^{\frac{1}{2}} ( x ^{\frac{1}{2}} ) ^{5} y ^{5}}{2 ^{\frac{1}{3}} ( x ^{2} ) ^{\frac{1}{3}} y ^{\frac{1}{3}} x ^{\frac{1}{3}}}

$(a^{m})^{n} = a^{m \cdot n}$

\frac{( 2 ^{3 \cdot \frac{1}{2}} ) ( x ^{3 \cdot \frac{1}{2}} ) ( x ^{\frac{1}{2} \cdot 5} ) y ^{5}}{2 ^{\frac{1}{3}} ( x ^{2 \cdot \frac{1}{3}} ) y ^{\frac{1}{3}} x ^{\frac{1}{3}}}

▼

Multiply

MoveLeftFacToNumOne

$a \cdot \frac{1}{b} = \frac{a}{b}$

\frac{2 ^{\frac{3}{2}} x ^{\frac{3}{2}} x ^{\frac{1}{2} \cdot 5} y ^{5}}{2 ^{\frac{1}{3}} x ^{\frac{2}{3}} y ^{\frac{1}{3}} x ^{\frac{1}{3}}}

MoveRightFacToNumOne

$\frac{1}{b} \cdot a = \frac{a}{b}$

\frac{2 ^{\frac{3}{2}} x ^{\frac{3}{2}} x ^{\frac{5}{2}} y ^{5}}{2 ^{\frac{1}{3}} x ^{\frac{2}{3}} y ^{\frac{1}{3}} x ^{\frac{1}{3}}}

CommutativePropMult

Commutative Property of Multiplication

\frac{2 ^{\frac{3}{2}} x ^{\frac{3}{2}} x ^{\frac{5}{2}} y ^{5}}{2 ^{\frac{1}{3}} x ^{\frac{2}{3}} x ^{\frac{1}{3}} y ^{\frac{1}{3}}}

$a^{m} \cdot a^{n} = a^{m + n}$

\frac{2 ^{\frac{3}{2}} ( x ^{\frac{3}{2} + \frac{5}{2}} ) y ^{5}}{2 ^{\frac{1}{3}} ( x ^{\frac{2}{3} + \frac{1}{3}} ) y ^{\frac{1}{3}}}

▼

Add terms

Add fractions

\frac{2 ^{\frac{3}{2}} x ^{\frac{8}{2}} y ^{5}}{2 ^{\frac{1}{3}} x ^{\frac{3}{3}} y ^{\frac{1}{3}}}

Calculate quotient

\frac{2 ^{\frac{3}{2}} x ^{4} y ^{5}}{2 ^{\frac{1}{3}} x ^{1} y ^{\frac{1}{3}}}

Write as a product of fractions

\frac{2 ^{\frac{3}{2}}}{2 ^{\frac{1}{3}}} \cdot \frac{x ^{4}}{x ^{1}} \cdot \frac{y ^{5}}{y ^{\frac{1}{3}}}

$\frac{a ^{m}}{a ^{n}} = a^{m - n}$

(2^{\frac{3}{2} - \frac{1}{3}}) (x^{4 - 1}) (y^{5 - \frac{1}{3}})

▼

Subtract terms

Subtract terms

(2^{\frac{3}{2} - \frac{1}{3}}) x^{3} (y^{5 - \frac{1}{3}})

$\frac{a}{b} = \frac{a \cdot 3}{b \cdot 3}$

(2^{\frac{9}{6} - \frac{1}{3}}) x^{3} (y^{5 - \frac{1}{3}})

$\frac{a}{b} = \frac{a \cdot 2}{b \cdot 2}$

(2^{\frac{9}{6} - \frac{2}{6}}) x^{3} (y^{5 - \frac{1}{3}})

$a = \frac{3 \cdot a}{3}$

(2^{\frac{9}{6} - \frac{2}{6}}) x^{3} (y^{\frac{1 5}{3} - \frac{1}{3}})

Subtract fractions

2^{\frac{7}{6}} x^{3} y^{\frac{1 4}{3}}

Extending the Properties of Exponents to Rational Exponents

Exercises

Calculate the value of the numeric expression without using a calculator. Write the answer in exact form.

\frac{4 2 5 \cdot 4 2 5 \cdot 4 2 5}{4 2 5 + 4 2 5 + 4 2 5}

We cannot calculate sqrt(25) without a calculator and we cannot reduce the fraction in its current form. However, we have an addition of three sqrt(25)-terms in the denominator. Therefore, we can start by rewriting this sum as a product.

Now, note that sqrt(25) is a factor of both the numerator and denominator. This means that we can cancel out common factors.

Let's now rewrite the radicals in the numerator as powers with rational exponents.

Next, we can use the Product of Powers Property for Rational Exponents. This property states that if we have a product of two powers with rational exponents and the same base, we can add the exponents and leave the base unchanged.

Finally, we can express the only power with a rational exponent as a radical expression, and evaluate.

The exact answer is 53.

Calculate the values of the expressions without using a calculator.

3 + 9^{\frac{1}{2}} - 3^{\frac{1}{2}} \cdot 3^{\frac{1}{2}}

2 - 8^{\frac{1}{3}} + 4^{\frac{1}{3}} \cdot 4^{\frac{2}{3}}

We will start by using the Product of Powers Property for Rational Exponents. This property states that if we have a product of two powers with the same base, we can add the exponents and leave the base unchanged. Then, we will express 9^(12) as a radical and evaluate this expression. Finally, we will add and subtract the terms. Let's do it!

We found that the given numeric expression is equal to 3.

Like in Part A, we will start by using the Product of Powers Property for Rational Exponents. Then, we will express 8^(13) as a radical and evaluate this radical expression. Finally, we will add and subtract the terms.

We found that the given expression is equal to 4.

The radius

r

of a cone's base is given by the following formula.

r = (\frac{3 V}{π h})^{\frac{1}{2}}

Here,

V

and

h

are the volume and the height of the cone, respectively. Find the radius of a cone with a volume of

7

cubic centimeters and a height of

5

centimeters. Use

π \approx 3.14

and approximate the answer to the nearest centimeter.

Let's start by drawing a cone with a volume of 7 cubic centimeters and a height of 5 centimeters.

Now we can substitute V=7 and h=5 into the given formula for the radius. Because π is a non-terminating and non-repeating decimal number, we will use 3.14 to approximate it.

Next, we will evaluate the obtained expression. To do so, we start by calculating the products in the numerator and the denominator.

Finally, we will use the Power of a Quotient Property for Rational Exponents. This property allows us to have the quotient of two powers instead of the power of a quotient.

We can now rewrite the powers with rational exponents as radical expressions and evaluate. Let's do it!

Rounded to the nearest centimeter, the radius of the base of the cone is 1 centimeter.

Extending the Properties of Exponents to Rational Exponents

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