Finite Math Examples

$\sqrt{\frac{z^{6}}{27 x^{9}}}$

Step 1

Rewrite $\frac{z^{6}}{27 x^{9}}$ as ${(\frac{z^{3}}{3 x^{4}})}^{2} \frac{1}{3 x}$ .

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Step 1.1

Factor the perfect power ${(z^{3})}^{2}$ out of $z^{6}$ .

$\sqrt{\frac{{(z^{3})}^{2} \cdot 1}{27 x^{9}}}$

Step 1.2

Factor the perfect power ${(3 x^{4})}^{2}$ out of $27 x^{9}$ .

$\sqrt{\frac{{(z^{3})}^{2} \cdot 1}{{(3 x^{4})}^{2} \cdot (3 x)}}$

Step 1.3

Rearrange the fraction $\frac{{(z^{3})}^{2} \cdot 1}{{(3 x^{4})}^{2} \cdot (3 x)}$ .

$\sqrt{{(\frac{z^{3}}{3 x^{4}})}^{2} \frac{1}{3 x}}$

Step 2

Pull terms out from under the radical.

$\frac{z^{3}}{3 x^{4}} \sqrt{\frac{1}{3 x}}$

Step 3

Rewrite $\sqrt{\frac{1}{3 x}}$ as $\frac{\sqrt{1}}{\sqrt{3 x}}$ .

$\frac{z^{3}}{3 x^{4}} \cdot \frac{\sqrt{1}}{\sqrt{3 x}}$

Step 4

Combine.

$\frac{z^{3} \sqrt{1}}{3 x^{4} \sqrt{3 x}}$

Step 5

Any root of $1$ is $1$ .

$\frac{z^{3} \cdot 1}{3 x^{4} \sqrt{3 x}}$

Step 6

Multiply $z^{3}$ by $1$ .

$\frac{z^{3}}{3 x^{4} \sqrt{3 x}}$

Step 7

Multiply $\frac{z^{3}}{3 x^{4} \sqrt{3 x}}$ by $\frac{\sqrt{3 x}}{\sqrt{3 x}}$ .

$\frac{z^{3}}{3 x^{4} \sqrt{3 x}} \cdot \frac{\sqrt{3 x}}{\sqrt{3 x}}$

Step 8

Combine and simplify the denominator.

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Step 8.1

Multiply $\frac{z^{3}}{3 x^{4} \sqrt{3 x}}$ by $\frac{\sqrt{3 x}}{\sqrt{3 x}}$ .

$\frac{z^{3} \sqrt{3 x}}{3 x^{4} \sqrt{3 x} \sqrt{3 x}}$

Step 8.2

Move $\sqrt{3 x}$ .

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

Step 8.3

Raise $\sqrt{3 x}$ to the power of $1$ .

$\frac{z^{3} \sqrt{3 x}}{3 x^{4} ({\sqrt{3 x}}^{1} \sqrt{3 x})}$

Step 8.4

Raise $\sqrt{3 x}$ to the power of $1$ .

$\frac{z^{3} \sqrt{3 x}}{3 x^{4} ({\sqrt{3 x}}^{1} {\sqrt{3 x}}^{1})}$

Step 8.5

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

$\frac{z^{3} \sqrt{3 x}}{3 x^{4} {\sqrt{3 x}}^{1 + 1}}$

Step 8.6

Add $1$ and $1$ .

$\frac{z^{3} \sqrt{3 x}}{3 x^{4} {\sqrt{3 x}}^{2}}$

Step 8.7

Rewrite ${\sqrt{3 x}}^{2}$ as $3 x$ .

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Step 8.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x}$ as ${(3 x)}^{\frac{1}{2}}$ .

$\frac{z^{3} \sqrt{3 x}}{3 x^{4} {({(3 x)}^{\frac{1}{2}})}^{2}}$

Step 8.7.2

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

$\frac{z^{3} \sqrt{3 x}}{3 x^{4} {(3 x)}^{\frac{1}{2} \cdot 2}}$

Step 8.7.3

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

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

Step 8.7.4

Cancel the common factor of $2$ .

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Step 8.7.4.1

Cancel the common factor.

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

Step 8.7.4.2

Rewrite the expression.

$\frac{z^{3} \sqrt{3 x}}{3 x^{4} {(3 x)}^{1}}$

Step 8.7.5

Simplify.

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

Step 9

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 9.1

Move $x$ .

$\frac{z^{3} \sqrt{3 x}}{3 (x \cdot x^{4}) \cdot 3}$

Step 9.2

Multiply $x$ by $x^{4}$ .

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Step 9.2.1

Raise $x$ to the power of $1$ .

$\frac{z^{3} \sqrt{3 x}}{3 (x^{1} x^{4}) \cdot 3}$

Step 9.2.2

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

$\frac{z^{3} \sqrt{3 x}}{3 x^{1 + 4} \cdot 3}$

Step 9.3

Add $1$ and $4$ .

$\frac{z^{3} \sqrt{3 x}}{3 x^{5} \cdot 3}$

Step 10

Multiply $3$ by $3$ .

$\frac{z^{3} \sqrt{3 x}}{9 x^{5}}$