Rational Exponents 3

Question 1 of 5

Simplify

$125^{-\frac{2}{3}}$

Write fractions as “a/b”

(1/25)

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Negative Power

$a^{-\color{#e65021}{n}}=\frac{1}{a^\color{#e65021}{n}}$

Fractional Powers

$a^{\frac{\color{#004ec4}{T}}{\color{#D800AD}{B}}}=(\sqrt[\color{#D800AD}{B}]{a})^{\color{#004ec4}{T}}$

First, use Negative Powers to simplify the expression.

$125^{-\color{#e65021}{\frac{2}{3}}}$

$=$

$\frac{1}{125^\color{#e65021}{\frac{2}{3}}}$

Next, simplify further using Fractional Powers.

$\frac{1}{125^{\frac{\color{#004ec4}{2}}{\color{#D800AD}{3}}}}$	$=$	$\frac{1}{(\sqrt[\color{#D800AD}{3}]{125})^{\color{#004ec4}{2}}}$
	$=$	$1/(5)^2$	$root (3)(125)=5$
	$=$	$1/(25)$	$5^2=25$

$1/(25)$

Question 2 of 5

Simplify

$[8^(2/3)]^(1/2)$

(2)

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Power of a Power

${(a^\color{#007DDC}{m})}^{\color{#9a00c7}{n}}=a^{\color{#007DDC}{m} \times \color{#9a00c7}{n}}$

Fractional Powers

$a^{\frac{\color{#004ec4}{T}}{\color{#D800AD}{B}}}=(\sqrt[\color{#D800AD}{B}]{a})^{\color{#004ec4}{T}}$

First, use Power of a Power to simplify the expression.

${[8^{\color{#007DDC}{\frac{2}{3}}})}^\color{#9a00c7}{\frac{1}{2}}$	$=$	$8^{{\color{#007DDC}{\frac{2}{3}}} \times {\color{#9a00c7}{\frac{1}{2}}}}$
	$=$	$8^{\frac{2}{6}}$
	$=$	$8^{\frac{1}{3}}$	Simplify

Next, simplify further using Fractional Powers.

$8^{\frac{\color{#004ec4}{1}}{\color{#D800AD}{3}}}$	$=$	$(\sqrt[\color{#D800AD}{3}]{8})^{\color{#004ec4}{1}}$
	$=$	$(2)^1$	$root (3)(8)=2$
	$=$	$2$	$2^1=2$

$2$

Question 3 of 5

Simplify

$(32x)^(2/5)$

1. $4x^(2/5)$
2. $(x^(2/5))/4$
3. $16x^2$
4. $4x^5$

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Power of a Power

${(a^\color{#007DDC}{m})}^{\color{#9a00c7}{n}}=a^{\color{#007DDC}{m} \times \color{#9a00c7}{n}}$

Fractional Powers

$a^{\frac{\color{#004ec4}{T}}{\color{#D800AD}{B}}}=(\sqrt[\color{#D800AD}{B}]{a})^{\color{#004ec4}{T}}$

First, use Power of a Power to simplify the expression.

$(32x)^\color{#9a00c7}{\frac{2}{5}}$

$=$

$32^{\color{#9a00c7}{\frac{2}{5}}} \times x^{\color{#9a00c7}{\frac{2}{5}}}$

Next, simplify further using Fractional Powers.

$32^{\frac{\color{#004ec4}{2}}{\color{#D800AD}{5}}} \times x^{\frac{2}{5}}$	$=$	$(\sqrt[\color{#D800AD}{5}]{32})^{\color{#004ec4}{2}} \times x^{\frac{2}{5}}$
	$=$	$(2)^2 xx x^(2/5)$	$root (5)(32)=2$
	$=$	$4x^(2/5)$	$2^2=4$

$4x^(2/5)$

Question 4 of 5

Simplify

$(64a^(12))^(4/3)$

1. $128a^3$
2. $256a^(16)$
3. $(16)/(a^4)$
4. $32a^3$

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Power of a Power

${(a^\color{#007DDC}{m})}^{\color{#9a00c7}{n}}=a^{\color{#007DDC}{m} \times \color{#9a00c7}{n}}$

Fractional Powers

$a^{\frac{\color{#004ec4}{T}}{\color{#D800AD}{B}}}=(\sqrt[\color{#D800AD}{B}]{a})^{\color{#004ec4}{T}}$

First, use Power of a Power to simplify the expression.

$(64a^{\color{#007DDC}{12}})^\color{#9a00c7}{\frac{4}{3}}$	$=$	$64^{\color{#9a00c7}{\frac{4}{3}}} \times a^{\color{#007DDC}{12} \times \color{#9a00c7}{\frac{4}{3}}}$
	$=$	$64^(4/3) xx a^((48)/3)$
	$=$	$64^(4/3) xx a^(16)$	Simplify

Next, simplify further using Fractional Powers.

$64^{\frac{\color{#004ec4}{4}}{\color{#D800AD}{3}}} \times a^{16}$	$=$	$(\sqrt[\color{#D800AD}{3}]{64})^{\color{#004ec4}{4}} \times a^{16}$
	$=$	$(4)^4 xx a^(16)$	$root (3)(64)=4$
	$=$	$256a^(16)$	$4^4=256$

$256a^(16)$

Question 5 of 5

Simplify

$(25w^(10))^(3/2)$

1. $w/(125)$
2. $5/(w^5)$
3. $25w^5$
4. $125w^(15)$

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Power of a Power

${(a^\color{#007DDC}{m})}^{\color{#9a00c7}{n}}=a^{\color{#007DDC}{m} \times \color{#9a00c7}{n}}$

Fractional Powers

$a^{\frac{\color{#004ec4}{T}}{\color{#D800AD}{B}}}=(\sqrt[\color{#D800AD}{B}]{a})^{\color{#004ec4}{T}}$

First, use Power of a Power to simplify the expression.

$(25w^{\color{#007DDC}{10}})^\color{#9a00c7}{\frac{3}{2}}$	$=$	$25^{\color{#9a00c7}{\frac{3}{2}}} \times w^{\color{#007DDC}{10} \times \color{#9a00c7}{\frac{3}{2}}}$
	$=$	$25^(3/2) xx w^((30)/2)$
	$=$	$25^(3/2) xx w^(15)$	Simplify

Next, simplify further using Fractional Powers.

$25^{\frac{\color{#004ec4}{3}}{\color{#D800AD}{2}}} \times w^{15}$	$=$	$(\sqrt[\color{#D800AD}{2}]{25})^{\color{#004ec4}{3}} \times w^{15}$
	$=$	$(5)^3 xx w^(15)$	$root (2)(25)=5$
	$=$	$125w^(15)$	$5^3=125$

$125w^(15)$

Rational Exponents 3

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Quiz summary

Information

1. Question

Hint

Nice Job!

Negative Power

Fractional Powers

2. Question

Hint

Correct!

Power of a Power

Fractional Powers

3. Question

Hint

Well Done!

Power of a Power

Fractional Powers

4. Question

Hint

Exceptional!

Power of a Power

Fractional Powers

5. Question

Hint

Great Work!

Power of a Power

Fractional Powers

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