Simplify the Radical Expression 5x^4y^2 square root of 4/25x^20y^30

$5 x^{4} y^{2} \sqrt{\frac{4}{25} x^{20} y^{30}}$

Combine $\frac{4}{25}$ and $x^{20}$ .

$5 x^{4} y^{2} \sqrt{\frac{4 x^{20}}{25} y^{30}}$

Combine $\frac{4 x^{20}}{25}$ and $y^{30}$ .

$5 x^{4} y^{2} \sqrt{\frac{4 x^{20} y^{30}}{25}}$

Rewrite $4 x^{20} y^{30}$ as ${(2 x^{10} y^{15})}^{2}$ .

$5 x^{4} y^{2} \sqrt{\frac{{(2 x^{10} y^{15})}^{2}}{25}}$

Rewrite $25$ as $5^{2}$ .

$5 x^{4} y^{2} \sqrt{\frac{{(2 x^{10} y^{15})}^{2}}{5^{2}}}$

Rewrite $\frac{{(2 x^{10} y^{15})}^{2}}{5^{2}}$ as ${(\frac{2 x^{10} y^{15}}{5})}^{2}$ .

$5 x^{4} y^{2} \sqrt{{(\frac{2 x^{10} y^{15}}{5})}^{2}}$

Pull terms out from under the radical.

$5 x^{4} y^{2} | \frac{2 x^{10} y^{15}}{5} |$

Remove non-negative terms from the absolute value.

$5 x^{4} y^{2} \frac{2 x^{10} | y^{15} |}{5}$

Cancel the common factor of $5$ .

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

$5 (x^{4} y^{2}) \frac{2 x^{10} | y^{15} |}{5}$

Cancel the common factor.

$5 (x^{4} y^{2}) \frac{2 x^{10} | y^{15} |}{5}$

Rewrite the expression.

$x^{4} y^{2} (2 x^{10} | y^{15} |)$

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

$x^{10 + 4} y^{2} (2 | y^{15} |)$

Add $10$ and $4$ .

$x^{14} y^{2} (2 | y^{15} |)$

Rewrite using the commutative property of multiplication.

$2 x^{14} y^{2} | y^{15} |$