Trigonometry Examples

$\frac{\sqrt{2}}{\sqrt{3 z}}$

Step 1

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

$\frac{\sqrt{2}}{\sqrt{3 z}} \cdot \frac{\sqrt{3 z}}{\sqrt{3 z}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

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

$\frac{\sqrt{2} \sqrt{3 z}}{\sqrt{3 z} \sqrt{3 z}}$

Step 2.2

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

$\frac{\sqrt{2} \sqrt{3 z}}{{\sqrt{3 z}}^{1} \sqrt{3 z}}$

Step 2.3

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

$\frac{\sqrt{2} \sqrt{3 z}}{{\sqrt{3 z}}^{1} {\sqrt{3 z}}^{1}}$

Step 2.4

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

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

Step 2.5

Add $1$ and $1$ .

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

Step 2.6

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

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Step 2.6.1

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

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

Step 2.6.2

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

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

Step 2.6.3

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

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

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

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

Step 2.6.4.2

Rewrite the expression.

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

Step 2.6.5

Simplify.

$\frac{\sqrt{2} \sqrt{3 z}}{3 z}$

Step 3

Simplify the numerator.

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Step 3.1

Combine using the product rule for radicals.

$\frac{\sqrt{2 (3 z)}}{3 z}$

Step 3.2

Multiply $2$ by $3$ .

$\frac{\sqrt{6 z}}{3 z}$