Simplify ( sixth root of x^7z^-2)^5

${(\sqrt[6]{x^{7} z^{- 2}})}^{5}$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${\sqrt[6]{x^{7} \frac{1}{z^{2}}}}^{5}$

Step 2

Combine $x^{7}$ and $\frac{1}{z^{2}}$ .

${\sqrt[6]{\frac{x^{7}}{z^{2}}}}^{5}$

Step 3

Rewrite $\sqrt[6]{\frac{x^{7}}{z^{2}}}$ as $\frac{\sqrt[6]{x^{7}}}{\sqrt[6]{z^{2}}}$ .

${(\frac{\sqrt[6]{x^{7}}}{\sqrt[6]{z^{2}}})}^{5}$

Step 4

Simplify the numerator.

Factor out $x^{6}$ .

${(\frac{\sqrt[6]{x^{6} x}}{\sqrt[6]{z^{2}}})}^{5}$

Pull terms out from under the radical.

${(\frac{x \sqrt[6]{x}}{\sqrt[6]{z^{2}}})}^{5}$

Step 5

Simplify the denominator.

Rewrite $\sqrt[6]{z^{2}}$ as $\sqrt[3]{\sqrt{z^{2}}}$ .

${(\frac{x \sqrt[6]{x}}{\sqrt[3]{\sqrt{z^{2}}}})}^{5}$

Pull terms out from under the radical, assuming positive real numbers.

${(\frac{x \sqrt[6]{x}}{\sqrt[3]{z}})}^{5}$

Step 6

Multiply $\frac{x \sqrt[6]{x}}{\sqrt[3]{z}}$ by $\frac{{\sqrt[3]{z}}^{2}}{{\sqrt[3]{z}}^{2}}$ .

${(\frac{x \sqrt[6]{x}}{\sqrt[3]{z}} \cdot \frac{{\sqrt[3]{z}}^{2}}{{\sqrt[3]{z}}^{2}})}^{5}$

Step 7

Combine and simplify the denominator.

Multiply $\frac{x \sqrt[6]{x}}{\sqrt[3]{z}}$ by $\frac{{\sqrt[3]{z}}^{2}}{{\sqrt[3]{z}}^{2}}$ .

${(\frac{x \sqrt[6]{x} {\sqrt[3]{z}}^{2}}{\sqrt[3]{z} {\sqrt[3]{z}}^{2}})}^{5}$

Raise $\sqrt[3]{z}$ to the power of $1$ .

${(\frac{x \sqrt[6]{x} {\sqrt[3]{z}}^{2}}{{\sqrt[3]{z}}^{1} {\sqrt[3]{z}}^{2}})}^{5}$

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

${(\frac{x \sqrt[6]{x} {\sqrt[3]{z}}^{2}}{{\sqrt[3]{z}}^{1 + 2}})}^{5}$

Add $1$ and $2$ .

${(\frac{x \sqrt[6]{x} {\sqrt[3]{z}}^{2}}{{\sqrt[3]{z}}^{3}})}^{5}$

Rewrite ${\sqrt[3]{z}}^{3}$ as $z$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{z}$ as $z^{\frac{1}{3}}$ .

${(\frac{x \sqrt[6]{x} {\sqrt[3]{z}}^{2}}{{(z^{\frac{1}{3}})}^{3}})}^{5}$

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

${(\frac{x \sqrt[6]{x} {\sqrt[3]{z}}^{2}}{z^{\frac{1}{3} \cdot 3}})}^{5}$

Combine $\frac{1}{3}$ and $3$ .

${(\frac{x \sqrt[6]{x} {\sqrt[3]{z}}^{2}}{z^{\frac{3}{3}}})}^{5}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(\frac{x \sqrt[6]{x} {\sqrt[3]{z}}^{2}}{z^{\frac{3}{3}}})}^{5}$

Rewrite the expression.

${(\frac{x \sqrt[6]{x} {\sqrt[3]{z}}^{2}}{z^{1}})}^{5}$

Simplify.

${(\frac{x \sqrt[6]{x} {\sqrt[3]{z}}^{2}}{z})}^{5}$

Step 8

Simplify the numerator.

Rewrite ${\sqrt[3]{z}}^{2}$ as $\sqrt[3]{z^{2}}$ .

${(\frac{x \sqrt[6]{x} \sqrt[3]{z^{2}}}{z})}^{5}$

Combine exponents.

Rewrite the expression using the least common index of $6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{z^{2}}$ as $z^{\frac{2}{3}}$ .

${(\frac{x (z^{\frac{2}{3}} \sqrt[6]{x})}{z})}^{5}$

Rewrite $z^{\frac{2}{3}}$ as $z^{\frac{4}{6}}$ .

${(\frac{x (z^{\frac{4}{6}} \sqrt[6]{x})}{z})}^{5}$

Rewrite $z^{\frac{4}{6}}$ as $\sqrt[6]{z^{4}}$ .

${(\frac{x (\sqrt[6]{z^{4}} \sqrt[6]{x})}{z})}^{5}$

Combine using the product rule for radicals.

${(\frac{x \sqrt[6]{z^{4} x}}{z})}^{5}$

Step 9

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

Apply the product rule to $\frac{x \sqrt[6]{z^{4} x}}{z}$ .

$\frac{{(x \sqrt[6]{z^{4} x})}^{5}}{z^{5}}$

Apply the product rule to $x \sqrt[6]{z^{4} x}$ .

$\frac{x^{5} {\sqrt[6]{z^{4} x}}^{5}}{z^{5}}$

Step 10

Simplify the numerator.

Rewrite ${\sqrt[6]{z^{4} x}}^{5}$ as $\sqrt[6]{{(z^{4} x)}^{5}}$ .

$\frac{x^{5} \sqrt[6]{{(z^{4} x)}^{5}}}{z^{5}}$

Apply the product rule to $z^{4} x$ .

$\frac{x^{5} \sqrt[6]{{(z^{4})}^{5} x^{5}}}{z^{5}}$

Multiply the exponents in ${(z^{4})}^{5}$ .

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

$\frac{x^{5} \sqrt[6]{z^{4 \cdot 5} x^{5}}}{z^{5}}$

Multiply $4$ by $5$ .

$\frac{x^{5} \sqrt[6]{z^{20} x^{5}}}{z^{5}}$

Rewrite $z^{20} x^{5}$ as ${(z^{3})}^{6} (z^{2} x^{5})$ .

Factor out $z^{18}$ .

$\frac{x^{5} \sqrt[6]{z^{18} z^{2} x^{5}}}{z^{5}}$

Rewrite $z^{18}$ as ${(z^{3})}^{6}$ .

$\frac{x^{5} \sqrt[6]{{(z^{3})}^{6} z^{2} x^{5}}}{z^{5}}$

Add parentheses.

$\frac{x^{5} \sqrt[6]{{(z^{3})}^{6} (z^{2} x^{5})}}{z^{5}}$

Pull terms out from under the radical.

$\frac{x^{5} z^{3} \sqrt[6]{z^{2} x^{5}}}{z^{5}}$

Step 11

Cancel the common factor of $z^{3}$ and $z^{5}$ .

Factor $z^{3}$ out of $x^{5} z^{3} \sqrt[6]{z^{2} x^{5}}$ .

$\frac{z^{3} (x^{5} \sqrt[6]{z^{2} x^{5}})}{z^{5}}$

Cancel the common factors.

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

$\frac{z^{3} (x^{5} \sqrt[6]{z^{2} x^{5}})}{z^{3} z^{2}}$

Cancel the common factor.

$\frac{z^{3} (x^{5} \sqrt[6]{z^{2} x^{5}})}{z^{3} z^{2}}$

Rewrite the expression.

$\frac{x^{5} \sqrt[6]{z^{2} x^{5}}}{z^{2}}$

Simplify ( sixth root of x^7z^-2)^5

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