Trigonometry Examples

$\frac{\sqrt{15 x y}}{3 \sqrt{10 x y^{3}}}$

Step 1

Combine $\sqrt{15 x y}$ and $\sqrt{10 x y^{3}}$ into a single radical.

$\frac{\sqrt{\frac{15 x y}{10 x y^{3}}}}{3}$

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

Cancel the common factor of $15$ and $10$ .

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Step 2.1.1

Factor $5$ out of $15 x y$ .

$\frac{\sqrt{\frac{5 (3 x y)}{10 x y^{3}}}}{3}$

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Factor $5$ out of $10 x y^{3}$ .

$\frac{\sqrt{\frac{5 (3 x y)}{5 (2 x y^{3})}}}{3}$

Step 2.1.2.2

Cancel the common factor.

$\frac{\sqrt{\frac{5 (3 x y)}{5 (2 x y^{3})}}}{3}$

Step 2.1.2.3

Rewrite the expression.

$\frac{\sqrt{\frac{3 x y}{2 x y^{3}}}}{3}$

Step 2.2

Cancel the common factor of $x$ .

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Step 2.2.1

Cancel the common factor.

$\frac{\sqrt{\frac{3 x y}{2 x y^{3}}}}{3}$

Step 2.2.2

Rewrite the expression.

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

Step 2.3

Cancel the common factor of $y$ and $y^{3}$ .

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Step 2.3.1

Factor $y$ out of $3 y$ .

$\frac{\sqrt{\frac{y \cdot 3}{2 y^{3}}}}{3}$

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

Factor $y$ out of $2 y^{3}$ .

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

Step 2.3.2.2

Cancel the common factor.

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

Step 2.3.2.3

Rewrite the expression.

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

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $\frac{3}{2 y^{2}}$ as ${(\frac{1}{y})}^{2} \frac{3}{2}$ .

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Step 3.1.1

Factor the perfect power $1^{2}$ out of $3$ .

$\frac{\sqrt{\frac{1^{2} \cdot 3}{2 y^{2}}}}{3}$

Step 3.1.2

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

$\frac{\sqrt{\frac{1^{2} \cdot 3}{y^{2} \cdot 2}}}{3}$

Step 3.1.3

Rearrange the fraction $\frac{1^{2} \cdot 3}{y^{2} \cdot 2}$ .

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

Step 3.2

Pull terms out from under the radical.

$\frac{\frac{1}{y} \sqrt{\frac{3}{2}}}{3}$

Step 3.3

Rewrite $\sqrt{\frac{3}{2}}$ as $\frac{\sqrt{3}}{\sqrt{2}}$ .

$\frac{\frac{1}{y} \cdot \frac{\sqrt{3}}{\sqrt{2}}}{3}$

Step 3.4

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

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

Step 3.5

Combine and simplify the denominator.

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Step 3.5.1

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

$\frac{\frac{1}{y} \cdot \frac{\sqrt{3} \sqrt{2}}{\sqrt{2} \sqrt{2}}}{3}$

Step 3.5.2

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

$\frac{\frac{1}{y} \cdot \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}}{3}$

Step 3.5.3

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

$\frac{\frac{1}{y} \cdot \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}}{3}$

Step 3.5.4

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

$\frac{\frac{1}{y} \cdot \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}}{3}$

Step 3.5.5

Add $1$ and $1$ .

$\frac{\frac{1}{y} \cdot \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{2}}}{3}$

Step 3.5.6

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

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Step 3.5.6.1

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

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

Step 3.5.6.2

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

$\frac{\frac{1}{y} \cdot \frac{\sqrt{3} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}}{3}$

Step 3.5.6.3

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

$\frac{\frac{1}{y} \cdot \frac{\sqrt{3} \sqrt{2}}{2^{\frac{2}{2}}}}{3}$

Step 3.5.6.4

Cancel the common factor of $2$ .

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Step 3.5.6.4.1

Cancel the common factor.

$\frac{\frac{1}{y} \cdot \frac{\sqrt{3} \sqrt{2}}{2^{\frac{2}{2}}}}{3}$

Step 3.5.6.4.2

Rewrite the expression.

$\frac{\frac{1}{y} \cdot \frac{\sqrt{3} \sqrt{2}}{2^{1}}}{3}$

Step 3.5.6.5

Evaluate the exponent.

$\frac{\frac{1}{y} \cdot \frac{\sqrt{3} \sqrt{2}}{2}}{3}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Combine using the product rule for radicals.

$\frac{\frac{1}{y} \cdot \frac{\sqrt{3 \cdot 2}}{2}}{3}$

Step 3.6.2

Multiply $3$ by $2$ .

$\frac{\frac{1}{y} \cdot \frac{\sqrt{6}}{2}}{3}$

Step 3.7

Multiply $\frac{1}{y}$ by $\frac{\sqrt{6}}{2}$ .

$\frac{\frac{\sqrt{6}}{y \cdot 2}}{3}$

Step 3.8

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

$\frac{\frac{\sqrt{6}}{2 y}}{3}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt{6}}{2 y} \cdot \frac{1}{3}$

Step 5

Multiply $\frac{\sqrt{6}}{2 y} \cdot \frac{1}{3}$ .

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Step 5.1

Multiply $\frac{\sqrt{6}}{2 y}$ by $\frac{1}{3}$ .

$\frac{\sqrt{6}}{2 y \cdot 3}$

Step 5.2

Multiply $3$ by $2$ .

$\frac{\sqrt{6}}{6 y}$