Simplify 2 cube root of 6x^4y*(3 cube root of 9x^5y^2)

$2 \sqrt[3]{6 x^{4} y} \cdot (3 \sqrt[3]{9 x^{5} y^{2}})$

Step 1

Rewrite $6 x^{4} y$ as $x^{3} \cdot (6 x y)$ .

Factor out $x^{3}$ .

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

Reorder $6$ and $x^{3}$ .

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

Add parentheses.

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

Add parentheses.

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

Step 2

Pull terms out from under the radical.

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

Step 3

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

Factor out $x^{3}$ .

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

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

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

Add parentheses.

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

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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

Step 5

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

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

Multiply $x$ by $x$ .

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

Step 6

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

Multiply $3$ by $2$ .

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

Combine using the product rule for radicals.

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

Multiply $6$ by $9$ .

$6 x^{2} \sqrt[3]{54 x^{2} y^{2} (x y)}$

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

$6 x^{2} \sqrt[3]{54 (x^{1} x^{2}) y^{2} y}$

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

$6 x^{2} \sqrt[3]{54 x^{1 + 2} y^{2} y}$

Add $1$ and $2$ .

$6 x^{2} \sqrt[3]{54 x^{3} y^{2} y}$

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

$6 x^{2} \sqrt[3]{54 x^{3} (y^{1} y^{2})}$

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

$6 x^{2} \sqrt[3]{54 x^{3} y^{1 + 2}}$

Add $1$ and $2$ .

$6 x^{2} \sqrt[3]{54 x^{3} y^{3}}$

Step 7

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

Factor $27$ out of $54$ .

$6 x^{2} \sqrt[3]{27 (2) x^{3} y^{3}}$

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

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

Move $2$ .

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

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

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

Step 8

Pull terms out from under the radical.

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

Step 9

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

Move $x$ .

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

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

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

$6 (x^{1} x^{2}) (3 y \sqrt[3]{2})$

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

$6 x^{1 + 2} (3 y \sqrt[3]{2})$

Add $1$ and $2$ .

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

Step 10

Multiply $3$ by $6$ .

$18 x^{3} y \sqrt[3]{2}$

Simplify 2 cube root of 6x^4y*(3 cube root of 9x^5y^2)

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