Simplify ((3x^5y^0z^8)/(x^7z^-2))^4

${(\frac{3 x^{5} y^{0} z^{8}}{x^{7} z^{- 2}})}^{4}$

Step 1

Move $z^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{3 x^{5} y^{0} z^{8} z^{2}}{x^{7}})}^{4}$

Step 2

Simplify the numerator.

Multiply $z^{8}$ by $z^{2}$ by adding the exponents.

Move $z^{2}$ .

${(\frac{3 x^{5} y^{0} (z^{2} z^{8})}{x^{7}})}^{4}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(\frac{3 x^{5} y^{0} z^{2 + 8}}{x^{7}})}^{4}$

Add $2$ and $8$ .

${(\frac{3 x^{5} y^{0} z^{10}}{x^{7}})}^{4}$

Simplify $3 x^{5} y^{0} z^{10}$ .

${(\frac{3 x^{5} z^{10}}{x^{7}})}^{4}$

Step 3

Cancel the common factor of $x^{5}$ and $x^{7}$ .

Factor $x^{5}$ out of $3 x^{5} z^{10}$ .

${(\frac{x^{5} (3 z^{10})}{x^{7}})}^{4}$

Cancel the common factors.

Factor $x^{5}$ out of $x^{7}$ .

${(\frac{x^{5} (3 z^{10})}{x^{5} x^{2}})}^{4}$

Cancel the common factor.

${(\frac{x^{5} (3 z^{10})}{x^{5} x^{2}})}^{4}$

Rewrite the expression.

${(\frac{3 z^{10}}{x^{2}})}^{4}$

Step 4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{3 z^{10}}{x^{2}}$ .

$\frac{{(3 z^{10})}^{4}}{{(x^{2})}^{4}}$

Apply the product rule to $3 z^{10}$ .

$\frac{3^{4} {(z^{10})}^{4}}{{(x^{2})}^{4}}$

Step 5

Simplify the numerator.

Raise $3$ to the power of $4$ .

$\frac{81 {(z^{10})}^{4}}{{(x^{2})}^{4}}$

Multiply the exponents in ${(z^{10})}^{4}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{81 z^{10 \cdot 4}}{{(x^{2})}^{4}}$

Multiply $10$ by $4$ .

$\frac{81 z^{40}}{{(x^{2})}^{4}}$

Step 6

Multiply the exponents in ${(x^{2})}^{4}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{81 z^{40}}{x^{2 \cdot 4}}$

Multiply $2$ by $4$ .

$\frac{81 z^{40}}{x^{8}}$

Simplify ((3x^5y^0z^8)/(x^7z^-2))^4

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