Powers of a Power 4

Question 1 of 4

Simplify

{(\frac{m^{2}}{3 x^{3}})}^{2}

$((m^2)/(3x^3))^2$

1. $\frac{m^{4}}{6 x^{6}}$ $(m^4)/(6x^6)$
2. $3 m x^{2}$ $3mx^2$
3. $\frac{3 m}{x^{6}}$ $(3m)/(x^6)$
4. $\frac{m^{4}}{9 x^{6}}$ $(m^4)/(9x^6)$

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Power of a Power

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

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

Simplify the expression by using the Power of a Power.

${\left(\frac{m^{\color{#007DDC}{2}}}{3x^{\color{#007DDC}{3}}}\right)}^\color{#9a00c7}{2}$	$=$ $=$	$\frac{(m^{\color{#007DDC}{2}})^{\color{#9a00c7}{2}}}{(3x^{\color{#007DDC}{3}})^{\color{#9a00c7}{2}}}$

	$=$ $=$	$\frac{m^{\color{#007DDC}{2} \times \color{#9a00c7}{2}}}{3^{\color{#9a00c7}{2}}x^{\color{#007DDC}{3} \times \color{#9a00c7}{2}}}$

	$=$ $=$	$\frac{m^{4}}{9 x^{6}}$ $(m^4)/(9x^6)$

\frac{m^{4}}{9 x^{6}}

$(m^4)/(9x^6)$

Question 2 of 4

Simplify

{(\frac{2 a^{3}}{5 x})}^{2}

$((2a^3)/(5x))^2$

1. $\frac{4 a^{6}}{25 x^{2}}$ $(4a^6)/(25x^2)$
2. $\frac{a^{6}}{10 x}$ $(a^6)/(10x)$
3. $\frac{a^{12}}{5 x}$ $(a^12)/(5x)$
4. $\frac{4 a^{5}}{10 x}$ $(4a^5)/(10x)$

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Power of a Power

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

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

Simplify the expression by using the Power of a Power.

${\left(\frac{2a^{\color{#007DDC}{3}}}{5x}\right)}^\color{#9a00c7}{2}$	$=$ $=$	$\frac{(2a^{\color{#007DDC}{3}})^{\color{#9a00c7}{2}}}{(5x)^{\color{#9a00c7}{2}}}$

	$=$ $=$	$\frac{2^{\color{#9a00c7}{2}}a^{\color{#007DDC}{3} \times \color{#9a00c7}{2}}}{5^{\color{#9a00c7}{2}}x^{\color{#9a00c7}{2}}}$

	$=$ $=$	$\frac{4 a^{6}}{25 x^{2}}$ $(4a^6)/(25x^2)$

\frac{4 a^{6}}{25 x^{2}}

$(4a^6)/(25x^2)$

Question 3 of 4

Simplify

\frac{{(2 x^{3})}^{2}}{8 x^{8} \div 2 x^{2}}

$((2x^3)^2)/(8x^8-:2x^2)$

(1)

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Power of a Power

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

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

First, use the Quotient of Powers to simplify the denominator.

$\frac{(2x^3)^2}{8\color{#00880A}{x}^8 \div 2\color{#00880A}{x}^2}$	$=$ $=$	$\frac{(2x^3)^2}{(8 \div 2) \times {\color{#00880A}{x}^{8-2}}}$

	$=$ $=$	$\frac{(2x^3)^2}{4 \times {\color{#00880A}{x}^6}}$

	$=$ $=$	$\frac{{(2 x^{3})}^{2}}{4 x^{6}}$ $((2x^3)^2)/(4x^6)$

Simplify the expression further by using the Power of a Power.

$\frac{(2x^{\color{#007DDC}{3}})^{\color{#9a00c7}{2}}}{4x^6}$	$=$ $=$	$\frac{2^{\color{#9a00c7}{2}}x^{\color{#007DDC}{3} \times \color{#9a00c7}{2}}}{4x^6}$

	$=$ $=$	$\frac{4 x^{6}}{4 x^{6}}$ $(4x^6)/(4x^6)$

	$=$ $=$	$1$ $1$

1

$1$

Question 4 of 4

Simplify

{(\frac{a \sqrt{b}}{c^{- 2}})}^{2} \div {(\frac{a^{2} c^{3}}{b})}^{4}

$((asqrtb)/(c^(-2)))^2-:((a^2 c^3)/b)^4$

1. $\frac{\sqrt{a b}}{c^{4}}$ $(sqrt(ab))/(c^4)$
2. $\frac{2 a b}{4 c b}$ $(2ab)/(4cb)$
3. $\frac{b^{5}}{a^{6} c^{8}}$ $(b^5)/(a^6 c^8)$
4. $\frac{b}{4 a c^{6}}$ $b/(4ac^6)$

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Power of a Power

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

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

The equation can be written as a multiplication by getting the reciprocal of the divisor.

{(\frac{a \sqrt{b}}{c^{- 2}})}^{2} \div {(\frac{a^{2} c^{3}}{b})}^{4}

$((asqrtb)/(c^(-2)))^2-:((a^2 c^3)/b)^4$

=

$=$

{(\frac{a \sqrt{b}}{c^{- 2}})}^{2} \times {(\frac{b}{a^{2} c^{3}})}^{4}

$((asqrtb)/(c^(-2)))^2xx(b/(a^2 c^3))^4$

Use Power of a Power to separate the numerator and denominator.

$\left(\frac{a \sqrt{b}}{c^{\color{#007DDC}{-2}}}\right)^{\color{#9a00c7}{2}} \times \left(\frac{b}{a^2c^3}\right)^4$	$=$	$\frac{a^{\color{#9a00c7}{2}} b^{\color{#007DDC}{\frac{1}{2}} \times \color{#9a00c7}{2}}}{c^{\color{#007DDC}{-2} \times \color{#9a00c7}{2}}} \times \left(\frac{b}{a^2c^3}\right)^4$	$sqrtb=b^(1/2)$

	$=$	$\frac{a^2b^1}{c^{-4}} \times \left(\frac{b}{a^{\color{#007DDC}{2}}c^{\color{#007DDC}{3}}}\right)^{\color{#9a00c7}{4}}$

	$=$	$\frac{a^2b}{c^{-4}} \times \frac{b^{\color{#9a00c7}{4}}}{a^{\color{#007DDC}{2} \times \color{#9a00c7}{4}}c^{\color{#007DDC}{3} \times \color{#9a00c7}{4}}}$

	$=$	$(a^2 b)/(c^(-4)) xx (b^4)/(a^8 c^(12))$

Finally, use both Product and Quotient of Powers to simplify the equation further.

$\frac{a^2\color{#00880A}{b}}{c^{-4}} \times \frac{\color{#00880A}{b}^4}{a^8 c^{12}}$	$=$	$\frac{a^2\color{#00880A}{b}^{1+4}}{c^{-4} \times a^8 c^{12}}$

	$=$	$\frac{a^2 b^5}{\color{#00880A}{c}^{-4} \times a^8 \color{#00880A}{c}^{12}}$

	$=$	$\frac{a^2 b^5}{a^8 \color{#00880A}{c}^{-4+12}}$

	$=$	$\frac{a^2 b^5 \div \color{#CC0000}{a^2}}{a^8 c^8 \div \color{#CC0000}{a^2}}$	Divide the numerator and the denominator by $a^2$

	$=$	$\frac{\color{#00880A}{a}^{2-2} b^5}{\color{#00880A}{a}^{8-2} c^8}$

	$=$	$\frac{b^5}{\color{#00880A}{a}^6 c^8}$

$(b^5)/(a^6 c^8)$

Powers of a Power 4

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