Simplify square root of (x^5y^2)/(125xy)

$\sqrt{\frac{x^{5} y^{2}}{125 x y}}$

Step 1

Reduce the expression $\frac{x^{5} y^{2}}{125 x y}$ by cancelling the common factors.

Factor $x$ out of $x^{5} y^{2}$ .

$\sqrt{\frac{x (x^{4} y^{2})}{125 x y}}$

Factor $x$ out of $125 x y$ .

$\sqrt{\frac{x (x^{4} y^{2})}{x (125 y)}}$

Cancel the common factor.

$\sqrt{\frac{x (x^{4} y^{2})}{x (125 y)}}$

Rewrite the expression.

$\sqrt{\frac{x^{4} y^{2}}{125 y}}$

Step 2

Cancel the common factor of $y^{2}$ and $y$ .

Factor $y$ out of $x^{4} y^{2}$ .

$\sqrt{\frac{y (x^{4} y)}{125 y}}$

Cancel the common factors.

Factor $y$ out of $125 y$ .

$\sqrt{\frac{y (x^{4} y)}{y \cdot 125}}$

Cancel the common factor.

$\sqrt{\frac{y (x^{4} y)}{y \cdot 125}}$

Rewrite the expression.

$\sqrt{\frac{x^{4} y}{125}}$

Step 3

Rewrite $\frac{x^{4} y}{125}$ as ${(\frac{x^{2}}{5})}^{2} \frac{y}{5}$ .

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

$\sqrt{\frac{{(x^{2})}^{2} y}{125}}$

Factor the perfect power $5^{2}$ out of $125$ .

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

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

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

Step 4

Pull terms out from under the radical.

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

Step 5

Rewrite $\sqrt{\frac{y}{5}}$ as $\frac{\sqrt{y}}{\sqrt{5}}$ .

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

Step 6

Combine.

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

Step 7

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

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

Step 8

Combine and simplify the denominator.

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

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

Move $\sqrt{5}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 9

Combine using the product rule for radicals.

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

Step 10

Multiply $5$ by $5$ .

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

Simplify square root of (x^5y^2)/(125xy)

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