Simplify x^2y cube root of 54x^8y^7

$x^{2} y \sqrt[3]{54 x^{8} y^{7}}$

Step 1

Rewrite $54 x^{8} y^{7}$ as ${(3 x^{2} y^{2})}^{3} \cdot (2 x^{2} y)$ .

Factor $27$ out of $54$ .

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

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

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

Factor out $x^{6}$ .

$x^{2} y \sqrt[3]{3^{3} \cdot 2 (x^{6} x^{2}) y^{7}}$

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

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

Factor out $y^{6}$ .

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

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

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

Move $x^{2}$ .

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

Move $2$ .

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

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

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

Add parentheses.

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

Add parentheses.

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

Step 2

Pull terms out from under the radical.

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

Step 3

Rewrite using the commutative property of multiplication.

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

Step 4

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

Move $x^{2}$ .

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

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

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

Add $2$ and $2$ .

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

Step 5

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

Move $y^{2}$ .

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

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

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

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

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

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

Add $2$ and $1$ .

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

Simplify x^2y cube root of 54x^8y^7

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