Simplify ( square root of 5x^4y)/( square root of 2x^2y^3)

$\frac{\sqrt{5 x^{4} y}}{\sqrt{2 x^{2} y^{3}}}$

Step 1

Combine $\sqrt{5 x^{4} y}$ and $\sqrt{2 x^{2} y^{3}}$ into a single radical.

$\sqrt{\frac{5 x^{4} y}{2 x^{2} y^{3}}}$

Step 2

Reduce the expression $\frac{5 x^{4} y}{2 x^{2} y^{3}}$ by cancelling the common factors.

Factor $x^{2}$ out of $5 x^{4} y$ .

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

Cancel the common factor of $y$ and $y^{3}$ .

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

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

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

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

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

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

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

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

Step 5

Pull terms out from under the radical.

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

Step 6

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

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

Step 7

Multiply $\frac{\sqrt{5}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 8

Combine and simplify the denominator.

Multiply $\frac{\sqrt{5}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Raise $\sqrt{2}$ to the power of $1$ .

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

Raise $\sqrt{2}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 9

Simplify the numerator.

Combine using the product rule for radicals.

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

Multiply $5$ by $2$ .

$\frac{x}{y} \cdot \frac{\sqrt{10}}{2}$

Step 10

Combine fractions.

Multiply $\frac{x}{y}$ by $\frac{\sqrt{10}}{2}$ .

$\frac{x \sqrt{10}}{y \cdot 2}$

Move $2$ to the left of $y$ .

$\frac{x \sqrt{10}}{2 y}$

Simplify ( square root of 5x^4y)/( square root of 2x^2y^3)

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