Simplify 7x^3y cube root of 8x^5y^10

$7 x^{3} y \sqrt[3]{8 x^{5} y^{10}}$

Rewrite $8 x^{5} y^{10}$ as ${(2 x y^{3})}^{3} (x^{2} y)$ .

Rewrite $8$ as $2^{3}$ .

$7 x^{3} y \sqrt[3]{2^{3} x^{5} y^{10}}$

Factor out $x^{3}$ .

$7 x^{3} y \sqrt[3]{2^{3} (x^{3} x^{2}) y^{10}}$

Factor out $y^{9}$ .

$7 x^{3} y \sqrt[3]{2^{3} (x^{3} x^{2}) (y^{9} y)}$

Rewrite $y^{9}$ as ${(y^{3})}^{3}$ .

$7 x^{3} y \sqrt[3]{2^{3} (x^{3} x^{2}) ({(y^{3})}^{3} y)}$

Move $x^{2}$ .

$7 x^{3} y \sqrt[3]{2^{3} (x^{3}) {(y^{3})}^{3} x^{2} y}$

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

$7 x^{3} y \sqrt[3]{{(2 x y^{3})}^{3} x^{2} y}$

Add parentheses.

$7 x^{3} y \sqrt[3]{{(2 x y^{3})}^{3} (x^{2} y)}$

Pull terms out from under the radical.

$7 x^{3} y (2 x y^{3} \sqrt[3]{x^{2} y})$

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

Move $x$ .

$7 (x \cdot x^{3}) y (2 y^{3} \sqrt[3]{x^{2} y})$

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

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

$7 (x^{1} x^{3}) y (2 y^{3} \sqrt[3]{x^{2} y})$

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

$7 x^{1 + 3} y (2 y^{3} \sqrt[3]{x^{2} y})$

Add $1$ and $3$ .

$7 x^{4} y (2 y^{3} \sqrt[3]{x^{2} y})$

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

Move $y^{3}$ .

$7 x^{4} (y^{3} y) (2 \sqrt[3]{x^{2} y})$

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

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

$7 x^{4} (y^{3} y^{1}) (2 \sqrt[3]{x^{2} y})$

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

$7 x^{4} y^{3 + 1} (2 \sqrt[3]{x^{2} y})$

Add $3$ and $1$ .

$7 x^{4} y^{4} (2 \sqrt[3]{x^{2} y})$

Multiply $2$ by $7$ .

$14 x^{4} y^{4} \sqrt[3]{x^{2} y}$