Simplify cube root of 18x^3y^5* cube root of 72x^2y

$\sqrt[3]{18 x^{3} y^{5}} \cdot \sqrt[3]{72 x^{2} y}$

Step 1

Rewrite $18 x^{3} y^{5}$ as ${(x y)}^{3} \cdot (18 y^{2})$ .

Factor out $y^{3}$ .

$\sqrt[3]{18 x^{3} (y^{3} y^{2})} \cdot \sqrt[3]{72 x^{2} y}$

Move $18$ .

$\sqrt[3]{x^{3} y^{3} \cdot 18 y^{2}} \cdot \sqrt[3]{72 x^{2} y}$

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

$\sqrt[3]{{(x y)}^{3} \cdot 18 y^{2}} \cdot \sqrt[3]{72 x^{2} y}$

Add parentheses.

$\sqrt[3]{{(x y)}^{3} \cdot (18 y^{2})} \cdot \sqrt[3]{72 x^{2} y}$

Step 2

Pull terms out from under the radical.

$x y \sqrt[3]{18 y^{2}} \cdot \sqrt[3]{72 x^{2} y}$

Step 3

Rewrite $72 x^{2} y$ as $2^{3} \cdot (9 x^{2} y)$ .

Factor $8$ out of $72$ .

$x y \sqrt[3]{18 y^{2}} \cdot \sqrt[3]{8 (9) x^{2} y}$

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

$x y \sqrt[3]{18 y^{2}} \cdot \sqrt[3]{2^{3} \cdot 9 x^{2} y}$

Add parentheses.

$x y \sqrt[3]{18 y^{2}} \cdot \sqrt[3]{2^{3} \cdot 9 (x^{2} y)}$

Add parentheses.

$x y \sqrt[3]{18 y^{2}} \cdot \sqrt[3]{2^{3} \cdot (9 x^{2} y)}$

Step 4

Pull terms out from under the radical.

$x y \sqrt[3]{18 y^{2}} \cdot (2 \sqrt[3]{9 x^{2} y})$

Step 5

Rewrite using the commutative property of multiplication.

$2 x y \sqrt[3]{18 y^{2}} \sqrt[3]{9 x^{2} y}$

Step 6

Multiply $2 x y \sqrt[3]{18 y^{2}} \sqrt[3]{9 x^{2} y}$ .

Combine using the product rule for radicals.

$2 x y \sqrt[3]{9 x^{2} y (18 y^{2})}$

Multiply $18$ by $9$ .

$2 x y \sqrt[3]{162 x^{2} y \cdot y^{2}}$

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

Move $y^{2}$ .

$2 x y \sqrt[3]{162 x^{2} (y^{2} y)}$

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

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

$2 x y \sqrt[3]{162 x^{2} (y^{2} y^{1})}$

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

$2 x y \sqrt[3]{162 x^{2} y^{2 + 1}}$

Add $2$ and $1$ .

$2 x y \sqrt[3]{162 x^{2} y^{3}}$

Step 7

Rewrite $162 x^{2} y^{3}$ as ${(3 y)}^{3} \cdot (6 x^{2})$ .

Factor $27$ out of $162$ .

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

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

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

Move $x^{2}$ .

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

Move $6$ .

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

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

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

Add parentheses.

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

Step 8

Pull terms out from under the radical.

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

Step 9

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

Move $y$ .

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

Multiply $y$ by $y$ .

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

Step 10

Multiply $3$ by $2$ .

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

Simplify cube root of 18x^3y^5* cube root of 72x^2y

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