Simplify 8x^2y^2 cube root of 135x^3y^7

$8 x^{2} y^{2} \sqrt[3]{135 x^{3} y^{7}}$

Step 1

Rewrite $135 x^{3} y^{7}$ as ${(3 x y^{2})}^{3} \cdot (5 y)$ .

Factor $27$ out of $135$ .

$8 x^{2} y^{2} \sqrt[3]{27 (5) x^{3} y^{7}}$

Rewrite $27$ as $3^{3}$ .

$8 x^{2} y^{2} \sqrt[3]{3^{3} \cdot 5 x^{3} y^{7}}$

Factor out $y^{6}$ .

$8 x^{2} y^{2} \sqrt[3]{3^{3} \cdot 5 x^{3} (y^{6} y)}$

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

$8 x^{2} y^{2} \sqrt[3]{3^{3} \cdot 5 x^{3} ({(y^{2})}^{3} y)}$

Move $5$ .

$8 x^{2} y^{2} \sqrt[3]{3^{3} x^{3} {(y^{2})}^{3} \cdot 5 y}$

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

$8 x^{2} y^{2} \sqrt[3]{{(3 x y^{2})}^{3} \cdot 5 y}$

Add parentheses.

$8 x^{2} y^{2} \sqrt[3]{{(3 x y^{2})}^{3} \cdot (5 y)}$

Step 2

Pull terms out from under the radical.

$8 x^{2} y^{2} (3 x y^{2} \sqrt[3]{5 y})$

Step 3

Multiply $x^{2}$ by $x$ by adding the exponents.

Move $x$ .

$8 (x \cdot x^{2}) y^{2} (3 y^{2} \sqrt[3]{5 y})$

Multiply $x$ by $x^{2}$ .

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

$8 (x^{1} x^{2}) y^{2} (3 y^{2} \sqrt[3]{5 y})$

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

$8 x^{1 + 2} y^{2} (3 y^{2} \sqrt[3]{5 y})$

Add $1$ and $2$ .

$8 x^{3} y^{2} (3 y^{2} \sqrt[3]{5 y})$

Step 4

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

Move $y^{2}$ .

$8 x^{3} (y^{2} y^{2}) (3 \sqrt[3]{5 y})$

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

$8 x^{3} y^{2 + 2} (3 \sqrt[3]{5 y})$

Add $2$ and $2$ .

$8 x^{3} y^{4} (3 \sqrt[3]{5 y})$

Step 5

Multiply $3$ by $8$ .

$24 x^{3} y^{4} \sqrt[3]{5 y}$