Algebra Examples

${(\sqrt{z \sqrt{y \sqrt{x}}})}^{8}$

Step 1

Rewrite ${\sqrt{z \sqrt{y \sqrt{x}}}}^{8}$ as ${(z \sqrt{y \sqrt{x}})}^{4}$ .

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{z \sqrt{y \sqrt{x}}}$ as ${(z \sqrt{y \sqrt{x}})}^{\frac{1}{2}}$ .

${({(z \sqrt{y \sqrt{x}})}^{\frac{1}{2}})}^{8}$

Step 1.2

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

${(z \sqrt{y \sqrt{x}})}^{\frac{1}{2} \cdot 8}$

Step 1.3

Combine $\frac{1}{2}$ and $8$ .

${(z \sqrt{y \sqrt{x}})}^{\frac{8}{2}}$

Step 1.4

Cancel the common factor of $8$ and $2$ .

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Step 1.4.1

Factor $2$ out of $8$ .

${(z \sqrt{y \sqrt{x}})}^{\frac{2 \cdot 4}{2}}$

Step 1.4.2

Cancel the common factors.

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Step 1.4.2.1

Factor $2$ out of $2$ .

${(z \sqrt{y \sqrt{x}})}^{\frac{2 \cdot 4}{2 (1)}}$

Step 1.4.2.2

Cancel the common factor.

${(z \sqrt{y \sqrt{x}})}^{\frac{2 \cdot 4}{2 \cdot 1}}$

Step 1.4.2.3

Rewrite the expression.

${(z \sqrt{y \sqrt{x}})}^{\frac{4}{1}}$

Step 1.4.2.4

Divide $4$ by $1$ .

${(z \sqrt{y \sqrt{x}})}^{4}$

Step 2

Apply the product rule to $z \sqrt{y \sqrt{x}}$ .

$z^{4} {\sqrt{y \sqrt{x}}}^{4}$

Step 3

Rewrite ${\sqrt{y \sqrt{x}}}^{4}$ as ${(y \sqrt{x})}^{2}$ .

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y \sqrt{x}}$ as ${(y \sqrt{x})}^{\frac{1}{2}}$ .

$z^{4} {({(y \sqrt{x})}^{\frac{1}{2}})}^{4}$

Step 3.2

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

$z^{4} {(y \sqrt{x})}^{\frac{1}{2} \cdot 4}$

Step 3.3

Combine $\frac{1}{2}$ and $4$ .

$z^{4} {(y \sqrt{x})}^{\frac{4}{2}}$

Step 3.4

Cancel the common factor of $4$ and $2$ .

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Step 3.4.1

Factor $2$ out of $4$ .

$z^{4} {(y \sqrt{x})}^{\frac{2 \cdot 2}{2}}$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor $2$ out of $2$ .

$z^{4} {(y \sqrt{x})}^{\frac{2 \cdot 2}{2 (1)}}$

Step 3.4.2.2

Cancel the common factor.

$z^{4} {(y \sqrt{x})}^{\frac{2 \cdot 2}{2 \cdot 1}}$

Step 3.4.2.3

Rewrite the expression.

$z^{4} {(y \sqrt{x})}^{\frac{2}{1}}$

Step 3.4.2.4

Divide $2$ by $1$ .

$z^{4} {(y \sqrt{x})}^{2}$

Step 4

Apply the product rule to $y \sqrt{x}$ .

$z^{4} (y^{2} {\sqrt{x}}^{2})$

Step 5

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$z^{4} (y^{2} {(x^{\frac{1}{2}})}^{2})$

Step 5.2

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

$z^{4} (y^{2} x^{\frac{1}{2} \cdot 2})$

Step 5.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 5.4

Cancel the common factor of $2$ .

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Step 5.4.1

Cancel the common factor.

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

Step 5.4.2

Rewrite the expression.

$z^{4} (y^{2} x^{1})$

Step 5.5

Simplify.

$z^{4} (y^{2} x)$

$z^{4} y^{2} x$