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

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

Step 1

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

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

Step 2

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

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

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

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

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

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

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

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

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

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

Step 6

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

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

Step 7

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 8

Simplify the numerator.

Combine using the product rule for radicals.

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

Multiply $5$ by $2$ .

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

Step 9

Combine $| \frac{x}{y} |$ and $\frac{\sqrt{10 y}}{2 y}$ .

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

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

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