Powers of a Power 4

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Question 1 of 4

Simplify

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

`(m^4)/(9x^6)`
`(3m)/(x^6)`
`(m^4)/(6x^6)`
`3mx^2`

Question 2 of 4

Simplify

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

`(4a^6)/(25x^2)`
`(a^6)/(10x)`
`(a^12)/(5x)`
`(4a^5)/(10x)`

Question 3 of 4

Simplify

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

Question 4 of 4

Simplify

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

`(b^5)/(a^6 c^8)`
`(2ab)/(4cb)`
`(sqrt(ab))/(c^4)`
`b/(4ac^6)`

Incorrect

Power of a Power

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

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

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

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

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

	`=`	`((2x^3)^2)/(4x^6)`

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

	`=`	`(4x^6)/(4x^6)`

	`=`	`1`

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Quiz summary

Information

1. Question

Hint

Exceptional!

Power of a Power

2. Question

Hint

Fantastic!

Power of a Power

3. Question

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Keep Going!

Power of a Power

4. Question

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Excellent!

Power of a Power

Quizzes

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

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

	`=`	`(m^4)/(9x^6)`

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

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

	`=`	`(4a^6)/(25x^2)`