Trigonometry Examples

$\frac{\sqrt{125}}{\sqrt{5 x^{9}}}$

Step 1

Combine $\sqrt{125}$ and $\sqrt{5 x^{9}}$ into a single radical.

$\sqrt{\frac{125}{5 x^{9}}}$

Step 2

Reduce the expression $\frac{125}{5 x^{9}}$ by cancelling the common factors.

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Step 2.1

Factor $5$ out of $125$ .

$\sqrt{\frac{5 \cdot 25}{5 x^{9}}}$

Step 2.2

Factor $5$ out of $5 x^{9}$ .

$\sqrt{\frac{5 \cdot 25}{5 (x^{9})}}$

Step 2.3

Cancel the common factor.

$\sqrt{\frac{5 \cdot 25}{5 x^{9}}}$

Step 2.4

Rewrite the expression.

$\sqrt{\frac{25}{x^{9}}}$

Step 3

Rewrite $\frac{25}{x^{9}}$ as ${(\frac{5}{x^{4}})}^{2} \frac{1}{x}$ .

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Step 3.1

Factor the perfect power $5^{2}$ out of $25$ .

$\sqrt{\frac{5^{2} \cdot 1}{x^{9}}}$

Step 3.2

Factor the perfect power ${(x^{4})}^{2}$ out of $x^{9}$ .

$\sqrt{\frac{5^{2} \cdot 1}{{(x^{4})}^{2} x}}$

Step 3.3

Rearrange the fraction $\frac{5^{2} \cdot 1}{{(x^{4})}^{2} x}$ .

$\sqrt{{(\frac{5}{x^{4}})}^{2} \frac{1}{x}}$

Step 4

Pull terms out from under the radical.

$\frac{5}{x^{4}} \sqrt{\frac{1}{x}}$

Step 5

Rewrite $\sqrt{\frac{1}{x}}$ as $\frac{\sqrt{1}}{\sqrt{x}}$ .

$\frac{5}{x^{4}} \cdot \frac{\sqrt{1}}{\sqrt{x}}$

Step 6

Combine.

$\frac{5 \sqrt{1}}{x^{4} \sqrt{x}}$

Step 7

Any root of $1$ is $1$ .

$\frac{5 \cdot 1}{x^{4} \sqrt{x}}$

Step 8

Multiply $5$ by $1$ .

$\frac{5}{x^{4} \sqrt{x}}$

Step 9

Multiply $\frac{5}{x^{4} \sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{5}{x^{4} \sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}$

Step 10

Combine and simplify the denominator.

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Step 10.1

Multiply $\frac{5}{x^{4} \sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{5 \sqrt{x}}{x^{4} \sqrt{x} \sqrt{x}}$

Step 10.2

Move $\sqrt{x}$ .

$\frac{5 \sqrt{x}}{x^{4} (\sqrt{x} \sqrt{x})}$

Step 10.3

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

$\frac{5 \sqrt{x}}{x^{4} ({\sqrt{x}}^{1} \sqrt{x})}$

Step 10.4

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

$\frac{5 \sqrt{x}}{x^{4} ({\sqrt{x}}^{1} {\sqrt{x}}^{1})}$

Step 10.5

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

$\frac{5 \sqrt{x}}{x^{4} {\sqrt{x}}^{1 + 1}}$

Step 10.6

Add $1$ and $1$ .

$\frac{5 \sqrt{x}}{x^{4} {\sqrt{x}}^{2}}$

Step 10.7

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

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Step 10.7.1

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

$\frac{5 \sqrt{x}}{x^{4} {(x^{\frac{1}{2}})}^{2}}$

Step 10.7.2

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

$\frac{5 \sqrt{x}}{x^{4} x^{\frac{1}{2} \cdot 2}}$

Step 10.7.3

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

$\frac{5 \sqrt{x}}{x^{4} x^{\frac{2}{2}}}$

Step 10.7.4

Cancel the common factor of $2$ .

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Step 10.7.4.1

Cancel the common factor.

$\frac{5 \sqrt{x}}{x^{4} x^{\frac{2}{2}}}$

Step 10.7.4.2

Rewrite the expression.

$\frac{5 \sqrt{x}}{x^{4} x^{1}}$

Step 10.7.5

Simplify.

$\frac{5 \sqrt{x}}{x^{4} x}$

Step 11

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 11.1

Multiply $x^{4}$ by $x$ .

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Step 11.1.1

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

$\frac{5 \sqrt{x}}{x^{4} x^{1}}$

Step 11.1.2

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

$\frac{5 \sqrt{x}}{x^{4 + 1}}$

Step 11.2

Add $4$ and $1$ .

$\frac{5 \sqrt{x}}{x^{5}}$