Simplify (3 cube root of 8x^2y^7)(5 cube root of 27x^5y^5)

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

Step 1

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

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

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

Factor out $y^{6}$ .

$3 \sqrt[3]{2^{3} x^{2} (y^{6} y)} (5 \sqrt[3]{27 x^{5} y^{5}})$

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

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

Move $x^{2}$ .

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

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

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

Add parentheses.

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

Step 2

Pull terms out from under the radical.

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

Step 3

Multiply $2$ by $3$ .

$6 (y^{2} \sqrt[3]{x^{2} y}) (5 \sqrt[3]{27 x^{5} y^{5}})$

Step 4

Rewrite $27 x^{5} y^{5}$ as ${(3 x y)}^{3} (x^{2} y^{2})$ .

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

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

Factor out $x^{3}$ .

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

Factor out $y^{3}$ .

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

Move $x^{2}$ .

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

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

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

Add parentheses.

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

Step 5

Pull terms out from under the radical.

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

Step 6

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

Move $y$ .

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

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

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

$6 (y^{1} y^{2}) \sqrt[3]{x^{2} y} (5 (3 x \sqrt[3]{x^{2} y^{2}}))$

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

$6 y^{1 + 2} \sqrt[3]{x^{2} y} (5 (3 x \sqrt[3]{x^{2} y^{2}}))$

Add $1$ and $2$ .

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

Step 7

Multiply $3$ by $5$ .

$6 y^{3} \sqrt[3]{x^{2} y} (15 (x \sqrt[3]{x^{2} y^{2}}))$

Step 8

Multiply $6 y^{3} \sqrt[3]{x^{2} y} (15 x \sqrt[3]{x^{2} y^{2}})$ .

Multiply $15$ by $6$ .

$90 y^{3} \sqrt[3]{x^{2} y} (x \sqrt[3]{x^{2} y^{2}})$

Combine using the product rule for radicals.

$90 y^{3} (x \sqrt[3]{x^{2} y (x^{2} y^{2})})$

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

$90 y^{3} (x \sqrt[3]{x^{2 + 2} y \cdot y^{2}})$

Add $2$ and $2$ .

$90 y^{3} (x \sqrt[3]{x^{4} y \cdot y^{2}})$

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

Move $y^{2}$ .

$90 y^{3} (x \sqrt[3]{x^{4} (y^{2} y)})$

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

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

$90 y^{3} (x \sqrt[3]{x^{4} (y^{2} y^{1})})$

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

$90 y^{3} (x \sqrt[3]{x^{4} y^{2 + 1}})$

Add $2$ and $1$ .

$90 y^{3} (x \sqrt[3]{x^{4} y^{3}})$

Step 9

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

Factor out $x^{3}$ .

$90 y^{3} (x \sqrt[3]{x^{3} x y^{3}})$

Move $x$ .

$90 y^{3} (x \sqrt[3]{x^{3} y^{3} x})$

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

$90 y^{3} (x \sqrt[3]{{(x y)}^{3} x})$

Step 10

Pull terms out from under the radical.

$90 y^{3} (x (x y \sqrt[3]{x}))$

Step 11

Multiply $y^{3}$ by $y$ by adding the exponents.

Move $y$ .

$90 (y \cdot y^{3}) (x (x \sqrt[3]{x}))$

Multiply $y$ by $y^{3}$ .

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

$90 (y^{1} y^{3}) (x (x \sqrt[3]{x}))$

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

$90 y^{1 + 3} (x (x \sqrt[3]{x}))$

Add $1$ and $3$ .

$90 y^{4} (x (x \sqrt[3]{x}))$

Step 12

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

Move $x$ .

$90 y^{4} (x \cdot x \sqrt[3]{x})$

Multiply $x$ by $x$ .

$90 y^{4} (x^{2} \sqrt[3]{x})$

$90 y^{4} x^{2} \sqrt[3]{x}$

Simplify (3 cube root of 8x^2y^7)(5 cube root of 27x^5y^5)

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