Simplify square root of (2x^3)/(50y^5)

$\sqrt{\frac{2 x^{3}}{50 y^{5}}}$

Step 1

Reduce the expression $\frac{2 x^{3}}{50 y^{5}}$ by cancelling the common factors.

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

$\sqrt{\frac{2 (x^{3})}{50 y^{5}}}$

Factor $2$ out of $50 y^{5}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 2

Rewrite $\frac{x^{3}}{25 y^{5}}$ as ${(\frac{x}{5 y^{2}})}^{2} \frac{x}{y}$ .

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

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

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

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

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

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

Step 3

Pull terms out from under the radical.

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

Step 4

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

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

Step 5

Combine.

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

Step 6

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

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

Step 7

Combine and simplify the denominator.

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

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

Move $\sqrt{y}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 8

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

Move $y$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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

Step 9

Combine using the product rule for radicals.

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

Simplify square root of (2x^3)/(50y^5)

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