Simplify xy^3 square root of 25x^7y^4

$x y^{3} \sqrt{25 x^{7} y^{4}}$

Rewrite $25 x^{7} y^{4}$ as ${(5 x^{3} y^{2})}^{2} x$ .

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

$x y^{3} \sqrt{5^{2} x^{7} y^{4}}$

Factor out $x^{6}$ .

$x y^{3} \sqrt{5^{2} (x^{6} x) y^{4}}$

Rewrite $x^{6}$ as ${(x^{3})}^{2}$ .

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

Rewrite $y^{4}$ as ${(y^{2})}^{2}$ .

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

Move $x$ .

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

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

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

Pull terms out from under the radical.

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

Rewrite using the commutative property of multiplication.

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

Multiply $x$ by $x^{3}$ by adding the exponents.

Move $x^{3}$ .

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

Multiply $x^{3}$ by $x$ .

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

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

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

$5 x^{3 + 1} y^{3} (y^{2} \sqrt{x})$

Add $3$ and $1$ .

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

Multiply $y^{3}$ by $y^{2}$ by adding the exponents.

Move $y^{2}$ .

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

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

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

Add $2$ and $3$ .

$5 x^{4} y^{5} \sqrt{x}$