Simplify ( cube root of 6x^2y)( cube root of 9xy^5)

$(\sqrt[3]{6 x^{2} y}) (\sqrt[3]{9 x y^{5}})$

Step 1

Rewrite $9 x y^{5}$ as $y^{3} \cdot (9 x y^{2})$ .

Factor out $y^{3}$ .

$\sqrt[3]{6 x^{2} y} \sqrt[3]{9 x (y^{3} y^{2})}$

Move $x$ .

$\sqrt[3]{6 x^{2} y} \sqrt[3]{9 y^{3} x y^{2}}$

Reorder $9$ and $y^{3}$ .

$\sqrt[3]{6 x^{2} y} \sqrt[3]{y^{3} \cdot 9 x y^{2}}$

Add parentheses.

$\sqrt[3]{6 x^{2} y} \sqrt[3]{y^{3} \cdot 9 (x y^{2})}$

Add parentheses.

$\sqrt[3]{6 x^{2} y} \sqrt[3]{y^{3} \cdot (9 x y^{2})}$

Step 2

Pull terms out from under the radical.

$\sqrt[3]{6 x^{2} y} (y \sqrt[3]{9 x y^{2}})$

Step 3

Multiply $\sqrt[3]{6 x^{2} y} (y \sqrt[3]{9 x y^{2}})$ .

Combine using the product rule for radicals.

$y \sqrt[3]{6 x^{2} y (9 x y^{2})}$

Multiply $9$ by $6$ .

$y \sqrt[3]{54 x^{2} y (x y^{2})}$

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

$y \sqrt[3]{54 (x^{1} x^{2}) y \cdot y^{2}}$

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

$y \sqrt[3]{54 x^{1 + 2} y \cdot y^{2}}$

Add $1$ and $2$ .

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

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

Move $y^{2}$ .

$y \sqrt[3]{54 x^{3} (y^{2} y)}$

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

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

$y \sqrt[3]{54 x^{3} (y^{2} y^{1})}$

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

$y \sqrt[3]{54 x^{3} y^{2 + 1}}$

Add $2$ and $1$ .

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

Step 4

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

Factor $27$ out of $54$ .

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

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

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

Move $2$ .

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

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

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

Step 5

Pull terms out from under the radical.

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

Step 6

Rewrite using the commutative property of multiplication.

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

Step 7

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

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

Multiply $y$ by $y$ .

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

Simplify ( cube root of 6x^2y)( cube root of 9xy^5)

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