Trigonometry Examples

$\frac{\sqrt{75 x^{5} y^{6}}}{\sqrt{36 x z^{5}}}$

Step 1

Combine $\sqrt{75 x^{5} y^{6}}$ and $\sqrt{36 x z^{5}}$ into a single radical.

$\sqrt{\frac{75 x^{5} y^{6}}{36 x z^{5}}}$

Step 2

Reduce the expression $\frac{75 x^{5} y^{6}}{36 x z^{5}}$ by cancelling the common factors.

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Step 2.1

Factor $3$ out of $75 x^{5} y^{6}$ .

$\sqrt{\frac{3 (25 x^{5} y^{6})}{36 x z^{5}}}$

Step 2.2

Factor $3$ out of $36 x z^{5}$ .

$\sqrt{\frac{3 (25 x^{5} y^{6})}{3 (12 x z^{5})}}$

Step 2.3

Cancel the common factor.

$\sqrt{\frac{3 (25 x^{5} y^{6})}{3 (12 x z^{5})}}$

Step 2.4

Rewrite the expression.

$\sqrt{\frac{25 x^{5} y^{6}}{12 x z^{5}}}$

Step 3

Cancel the common factor of $x^{5}$ and $x$ .

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Step 3.1

Factor $x$ out of $25 x^{5} y^{6}$ .

$\sqrt{\frac{x (25 x^{4} y^{6})}{12 x z^{5}}}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $x$ out of $12 x z^{5}$ .

$\sqrt{\frac{x (25 x^{4} y^{6})}{x (12 z^{5})}}$

Step 3.2.2

Cancel the common factor.

$\sqrt{\frac{x (25 x^{4} y^{6})}{x (12 z^{5})}}$

Step 3.2.3

Rewrite the expression.

$\sqrt{\frac{25 x^{4} y^{6}}{12 z^{5}}}$

Step 4

Rewrite $\frac{25 x^{4} y^{6}}{12 z^{5}}$ as ${(\frac{5 x^{2} y^{3}}{2 z^{2}})}^{2} \frac{1}{3 z}$ .

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Step 4.1

Factor the perfect power ${(5 x^{2} y^{3})}^{2}$ out of $25 x^{4} y^{6}$ .

$\sqrt{\frac{{(5 x^{2} y^{3})}^{2} \cdot 1}{12 z^{5}}}$

Step 4.2

Factor the perfect power ${(2 z^{2})}^{2}$ out of $12 z^{5}$ .

$\sqrt{\frac{{(5 x^{2} y^{3})}^{2} \cdot 1}{{(2 z^{2})}^{2} \cdot (3 z)}}$

Step 4.3

Rearrange the fraction $\frac{{(5 x^{2} y^{3})}^{2} \cdot 1}{{(2 z^{2})}^{2} \cdot (3 z)}$ .

$\sqrt{{(\frac{5 x^{2} y^{3}}{2 z^{2}})}^{2} \frac{1}{3 z}}$

Step 5

Pull terms out from under the radical.

$\frac{5 x^{2} y^{3}}{2 z^{2}} \sqrt{\frac{1}{3 z}}$

Step 6

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

$\frac{5 x^{2} y^{3}}{2 z^{2}} \cdot \frac{\sqrt{1}}{\sqrt{3 z}}$

Step 7

Combine.

$\frac{5 x^{2} y^{3} \sqrt{1}}{2 z^{2} \sqrt{3 z}}$

Step 8

Simplify the numerator.

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Step 8.1

Any root of $1$ is $1$ .

$\frac{5 x^{2} y^{3} \cdot 1}{2 z^{2} \sqrt{3 z}}$

Step 8.2

Multiply $5$ by $1$ .

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

Step 9

Multiply $\frac{5 x^{2} y^{3}}{2 z^{2} \sqrt{3 z}}$ by $\frac{\sqrt{3 z}}{\sqrt{3 z}}$ .

$\frac{5 x^{2} y^{3}}{2 z^{2} \sqrt{3 z}} \cdot \frac{\sqrt{3 z}}{\sqrt{3 z}}$

Step 10

Combine and simplify the denominator.

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Step 10.1

Multiply $\frac{5 x^{2} y^{3}}{2 z^{2} \sqrt{3 z}}$ by $\frac{\sqrt{3 z}}{\sqrt{3 z}}$ .

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

Step 10.2

Move $\sqrt{3 z}$ .

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

Step 10.3

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

$\frac{5 x^{2} y^{3} \sqrt{3 z}}{2 z^{2} ({\sqrt{3 z}}^{1} \sqrt{3 z})}$

Step 10.4

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

$\frac{5 x^{2} y^{3} \sqrt{3 z}}{2 z^{2} ({\sqrt{3 z}}^{1} {\sqrt{3 z}}^{1})}$

Step 10.5

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

$\frac{5 x^{2} y^{3} \sqrt{3 z}}{2 z^{2} {\sqrt{3 z}}^{1 + 1}}$

Step 10.6

Add $1$ and $1$ .

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

Step 10.7

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

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Step 10.7.1

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

$\frac{5 x^{2} y^{3} \sqrt{3 z}}{2 z^{2} {({(3 z)}^{\frac{1}{2}})}^{2}}$

Step 10.7.2

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

$\frac{5 x^{2} y^{3} \sqrt{3 z}}{2 z^{2} {(3 z)}^{\frac{1}{2} \cdot 2}}$

Step 10.7.3

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

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

Step 10.7.4

Cancel the common factor of $2$ .

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Step 10.7.4.1

Cancel the common factor.

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

Step 10.7.4.2

Rewrite the expression.

$\frac{5 x^{2} y^{3} \sqrt{3 z}}{2 z^{2} {(3 z)}^{1}}$

Step 10.7.5

Simplify.

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

Step 11

Multiply $z^{2}$ by $z$ by adding the exponents.

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Step 11.1

Move $z$ .

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

Step 11.2

Multiply $z$ by $z^{2}$ .

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Step 11.2.1

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

$\frac{5 x^{2} y^{3} \sqrt{3 z}}{2 (z^{1} z^{2}) \cdot 3}$

Step 11.2.2

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

$\frac{5 x^{2} y^{3} \sqrt{3 z}}{2 z^{1 + 2} \cdot 3}$

Step 11.3

Add $1$ and $2$ .

$\frac{5 x^{2} y^{3} \sqrt{3 z}}{2 z^{3} \cdot 3}$

Step 12

Multiply $3$ by $2$ .

$\frac{5 x^{2} y^{3} \sqrt{3 z}}{6 z^{3}}$