Pre-Algebra Examples

$- 3 \sqrt{\frac{z^{3}}{8 x^{6}}}$

Step 1

Rewrite $\frac{z^{3}}{8 x^{6}}$ as ${(\frac{z}{2 x^{3}})}^{2} \frac{z}{2}$ .

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Step 1.1

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

$- 3 \sqrt{\frac{z^{2} z}{8 x^{6}}}$

Step 1.2

Factor the perfect power ${(2 x^{3})}^{2}$ out of $8 x^{6}$ .

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

Step 1.3

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

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

Step 2

Pull terms out from under the radical.

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

Step 3

Rewrite $\sqrt{\frac{z}{2}}$ as $\frac{\sqrt{z}}{\sqrt{2}}$ .

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

Step 4

Combine.

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

Step 5

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

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

Step 6

Combine and simplify the denominator.

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Step 6.1

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

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

Step 6.2

Move $\sqrt{2}$ .

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

Step 6.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 6.4

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 6.5

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

$- 3 \frac{z \sqrt{z} \sqrt{2}}{2 x^{3} {\sqrt{2}}^{1 + 1}}$

Step 6.6

Add $1$ and $1$ .

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

Step 6.7

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

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Step 6.7.1

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

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

Step 6.7.2

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

$- 3 \frac{z \sqrt{z} \sqrt{2}}{2 x^{3} \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 6.7.3

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

$- 3 \frac{z \sqrt{z} \sqrt{2}}{2 x^{3} \cdot 2^{\frac{2}{2}}}$

Step 6.7.4

Cancel the common factor of $2$ .

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Step 6.7.4.1

Cancel the common factor.

$- 3 \frac{z \sqrt{z} \sqrt{2}}{2 x^{3} \cdot 2^{\frac{2}{2}}}$

Step 6.7.4.2

Rewrite the expression.

$- 3 \frac{z \sqrt{z} \sqrt{2}}{2 x^{3} \cdot 2^{1}}$

Step 6.7.5

Evaluate the exponent.

$- 3 \frac{z \sqrt{z} \sqrt{2}}{2 x^{3} \cdot 2}$

Step 7

Combine using the product rule for radicals.

$- 3 \frac{z \sqrt{2 z}}{2 x^{3} \cdot 2}$

Step 8

Multiply $2$ by $2$ .

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

Step 9

Combine $- 3$ and $\frac{z \sqrt{2 z}}{4 x^{3}}$ .

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

Step 10

Move the negative in front of the fraction.

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