Trigonometry Examples

$\frac{\sqrt{8 x}}{\sqrt{y}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $8 x$ as $2^{2} \cdot (2 x)$ .

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Step 1.1.1

Factor $4$ out of $8$ .

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

Step 1.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{2^{2} \cdot 2 x}}{\sqrt{y}}$

Step 1.1.3

Add parentheses.

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

Step 1.2

Pull terms out from under the radical.

$\frac{2 \sqrt{2 x}}{\sqrt{y}}$

Step 2

Multiply $\frac{2 \sqrt{2 x}}{\sqrt{y}}$ by $\frac{\sqrt{y}}{\sqrt{y}}$ .

$\frac{2 \sqrt{2 x}}{\sqrt{y}} \cdot \frac{\sqrt{y}}{\sqrt{y}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{2 \sqrt{2 x}}{\sqrt{y}}$ by $\frac{\sqrt{y}}{\sqrt{y}}$ .

$\frac{2 \sqrt{2 x} \sqrt{y}}{\sqrt{y} \sqrt{y}}$

Step 3.2

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

$\frac{2 \sqrt{2 x} \sqrt{y}}{{\sqrt{y}}^{1} \sqrt{y}}$

Step 3.3

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

$\frac{2 \sqrt{2 x} \sqrt{y}}{{\sqrt{y}}^{1} {\sqrt{y}}^{1}}$

Step 3.4

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

$\frac{2 \sqrt{2 x} \sqrt{y}}{{\sqrt{y}}^{1 + 1}}$

Step 3.5

Add $1$ and $1$ .

$\frac{2 \sqrt{2 x} \sqrt{y}}{{\sqrt{y}}^{2}}$

Step 3.6

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

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Step 3.6.1

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

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

Step 3.6.2

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

$\frac{2 \sqrt{2 x} \sqrt{y}}{y^{\frac{1}{2} \cdot 2}}$

Step 3.6.3

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

$\frac{2 \sqrt{2 x} \sqrt{y}}{y^{\frac{2}{2}}}$

Step 3.6.4

Cancel the common factor of $2$ .

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Step 3.6.4.1

Cancel the common factor.

$\frac{2 \sqrt{2 x} \sqrt{y}}{y^{\frac{2}{2}}}$

Step 3.6.4.2

Rewrite the expression.

$\frac{2 \sqrt{2 x} \sqrt{y}}{y^{1}}$

Step 3.6.5

Simplify.

$\frac{2 \sqrt{2 x} \sqrt{y}}{y}$

Step 4

Combine using the product rule for radicals.

$\frac{2 \sqrt{y \cdot 2 x}}{y}$

Step 5

Move $2$ to the left of $y$ .

$\frac{2 \sqrt{2 y x}}{y}$