Simplify the Radical Expression ( square root of 75x^5y)/( square root of 28xy^3)

$\frac{\sqrt{75 x^{5} y}}{\sqrt{28 x y^{3}}}$

Step 1

Combine $\sqrt{75 x^{5} y}$ and $\sqrt{28 x y^{3}}$ into a single radical.

$\sqrt{\frac{75 x^{5} y}{28 x y^{3}}}$

Step 2

Reduce the expression $\frac{75 x^{5} y}{28 x y^{3}}$ by cancelling the common factors.

Factor $x$ out of $75 x^{5} y$ .

$\sqrt{\frac{x (75 x^{4} y)}{28 x y^{3}}}$

Factor $x$ out of $28 x y^{3}$ .

$\sqrt{\frac{x (75 x^{4} y)}{x (28 y^{3})}}$

Cancel the common factor.

$\sqrt{\frac{x (75 x^{4} y)}{x (28 y^{3})}}$

Rewrite the expression.

$\sqrt{\frac{75 x^{4} y}{28 y^{3}}}$

Step 3

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

Factor $y$ out of $75 x^{4} y$ .

$\sqrt{\frac{y (75 x^{4})}{28 y^{3}}}$

Cancel the common factors.

Factor $y$ out of $28 y^{3}$ .

$\sqrt{\frac{y (75 x^{4})}{y (28 y^{2})}}$

Cancel the common factor.

$\sqrt{\frac{y (75 x^{4})}{y (28 y^{2})}}$

Rewrite the expression.

$\sqrt{\frac{75 x^{4}}{28 y^{2}}}$

Step 4

Rewrite $\frac{75 x^{4}}{28 y^{2}}$ as ${(\frac{5 x^{2}}{2 y})}^{2} \frac{3}{7}$ .

Factor the perfect power ${(5 x^{2})}^{2}$ out of $75 x^{4}$ .

$\sqrt{\frac{{(5 x^{2})}^{2} \cdot 3}{28 y^{2}}}$

Factor the perfect power ${(2 y)}^{2}$ out of $28 y^{2}$ .

$\sqrt{\frac{{(5 x^{2})}^{2} \cdot 3}{{(2 y)}^{2} \cdot 7}}$

Rearrange the fraction $\frac{{(5 x^{2})}^{2} \cdot 3}{{(2 y)}^{2} \cdot 7}$ .

$\sqrt{{(\frac{5 x^{2}}{2 y})}^{2} \frac{3}{7}}$

Step 5

Pull terms out from under the radical.

$| \frac{5 x^{2}}{2 y} | \sqrt{\frac{3}{7}}$

Step 6

Remove non-negative terms from the absolute value.

$\frac{5 x^{2}}{2 | y |} \sqrt{\frac{3}{7}}$

Step 7

Rewrite $\sqrt{\frac{3}{7}}$ as $\frac{\sqrt{3}}{\sqrt{7}}$ .

$\frac{5 x^{2}}{2 | y |} \cdot \frac{\sqrt{3}}{\sqrt{7}}$

Step 8

Combine.

$\frac{5 x^{2} \sqrt{3}}{2 | y | \sqrt{7}}$

Step 9

Multiply $\frac{5 x^{2} \sqrt{3}}{2 | y | \sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\frac{5 x^{2} \sqrt{3}}{2 | y | \sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 10

Combine and simplify the denominator.

Multiply $\frac{5 x^{2} \sqrt{3}}{2 | y | \sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\frac{5 x^{2} \sqrt{3} \sqrt{7}}{2 | y | \sqrt{7} \sqrt{7}}$

Move $\sqrt{7}$ .

$\frac{5 x^{2} \sqrt{3} \sqrt{7}}{2 | y | (\sqrt{7} \sqrt{7})}$

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{5 x^{2} \sqrt{3} \sqrt{7}}{2 | y | ({\sqrt{7}}^{1} \sqrt{7})}$

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{5 x^{2} \sqrt{3} \sqrt{7}}{2 | y | ({\sqrt{7}}^{1} {\sqrt{7}}^{1})}$

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

$\frac{5 x^{2} \sqrt{3} \sqrt{7}}{2 | y | {\sqrt{7}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{5 x^{2} \sqrt{3} \sqrt{7}}{2 | y | {\sqrt{7}}^{2}}$

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\frac{5 x^{2} \sqrt{3} \sqrt{7}}{2 | y | {(7^{\frac{1}{2}})}^{2}}$

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

$\frac{5 x^{2} \sqrt{3} \sqrt{7}}{2 | y | \cdot 7^{\frac{1}{2} \cdot 2}}$

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

$\frac{5 x^{2} \sqrt{3} \sqrt{7}}{2 | y | \cdot 7^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{5 x^{2} \sqrt{3} \sqrt{7}}{2 | y | \cdot 7^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{5 x^{2} \sqrt{3} \sqrt{7}}{2 | y | \cdot 7^{1}}$

Evaluate the exponent.

$\frac{5 x^{2} \sqrt{3} \sqrt{7}}{2 | y | \cdot 7}$

Step 11

Simplify the numerator.

Combine using the product rule for radicals.

$\frac{5 x^{2} \sqrt{7 \cdot 3}}{2 | y | \cdot 7}$

Multiply $7$ by $3$ .

$\frac{5 x^{2} \sqrt{21}}{2 | y | \cdot 7}$

Step 12

Multiply $7$ by $2$ .

$\frac{5 x^{2} \sqrt{21}}{14 | y |}$

Simplify the Radical Expression ( square root of 75x^5y)/( square root of 28xy^3)

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