Precalculus Examples

$\frac{25 x}{\sqrt{125 x^{3} y}}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $125 x^{3} y$ as ${(5 x)}^{2} \cdot (5 x y)$ .

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Step 1.1.1

Factor $25$ out of $125$ .

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

Step 1.1.2

Rewrite $25$ as $5^{2}$ .

$\frac{25 x}{\sqrt{5^{2} \cdot 5 x^{3} y}}$

Step 1.1.3

Factor out $x^{2}$ .

$\frac{25 x}{\sqrt{5^{2} \cdot 5 (x^{2} x) y}}$

Step 1.1.4

Move $5$ .

$\frac{25 x}{\sqrt{5^{2} x^{2} \cdot 5 x y}}$

Step 1.1.5

Rewrite $5^{2} x^{2}$ as ${(5 x)}^{2}$ .

$\frac{25 x}{\sqrt{{(5 x)}^{2} \cdot 5 x y}}$

Step 1.1.6

Add parentheses.

$\frac{25 x}{\sqrt{{(5 x)}^{2} \cdot 5 (x y)}}$

Step 1.1.7

Add parentheses.

$\frac{25 x}{\sqrt{{(5 x)}^{2} \cdot (5 x y)}}$

Step 1.2

Pull terms out from under the radical.

$\frac{25 x}{5 x \sqrt{5 x y}}$

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

Cancel the common factor of $25$ and $5$ .

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Step 2.1.1

Factor $5$ out of $25 x$ .

$\frac{5 (5 x)}{5 x \sqrt{5 x y}}$

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Factor $5$ out of $5 x \sqrt{5 x y}$ .

$\frac{5 (5 x)}{5 (x \sqrt{5 x y})}$

Step 2.1.2.2

Cancel the common factor.

$\frac{5 (5 x)}{5 (x \sqrt{5 x y})}$

Step 2.1.2.3

Rewrite the expression.

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

Step 2.2

Cancel the common factor of $x$ .

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Rewrite the expression.

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

Step 3

Multiply $\frac{5}{\sqrt{5 x y}}$ by $\frac{\sqrt{5 x y}}{\sqrt{5 x y}}$ .

$\frac{5}{\sqrt{5 x y}} \cdot \frac{\sqrt{5 x y}}{\sqrt{5 x y}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{5}{\sqrt{5 x y}}$ by $\frac{\sqrt{5 x y}}{\sqrt{5 x y}}$ .

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

Step 4.2

Raise $\sqrt{5 x y}$ to the power of $1$ .

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

Step 4.3

Raise $\sqrt{5 x y}$ to the power of $1$ .

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

Step 4.4

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

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

Step 4.5

Add $1$ and $1$ .

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

Step 4.6

Rewrite ${\sqrt{5 x y}}^{2}$ as $5 x y$ .

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Step 4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 x y}$ as ${(5 x y)}^{\frac{1}{2}}$ .

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

Step 4.6.2

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

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

Step 4.6.3

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

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

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

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

Step 4.6.4.2

Rewrite the expression.

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

Step 4.6.5

Simplify.

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

Step 5

Cancel the common factor of $5$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

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