Simplify ( cube root of 256y^8)( cube root of 625y^5)

$(\sqrt[3]{256 y^{8}}) (\sqrt[3]{625 y^{5}})$

Step 1

Rewrite $256 y^{8}$ as ${(4 y^{2})}^{3} \cdot (4 y^{2})$ .

Factor $64$ out of $256$ .

$\sqrt[3]{64 (4) y^{8}} \sqrt[3]{625 y^{5}}$

Rewrite $64$ as $4^{3}$ .

$\sqrt[3]{4^{3} \cdot 4 y^{8}} \sqrt[3]{625 y^{5}}$

Factor out $y^{6}$ .

$\sqrt[3]{4^{3} \cdot 4 (y^{6} y^{2})} \sqrt[3]{625 y^{5}}$

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

$\sqrt[3]{4^{3} \cdot 4 ({(y^{2})}^{3} y^{2})} \sqrt[3]{625 y^{5}}$

Move $4$ .

$\sqrt[3]{4^{3} {(y^{2})}^{3} \cdot 4 y^{2}} \sqrt[3]{625 y^{5}}$

Rewrite $4^{3} {(y^{2})}^{3}$ as ${(4 y^{2})}^{3}$ .

$\sqrt[3]{{(4 y^{2})}^{3} \cdot 4 y^{2}} \sqrt[3]{625 y^{5}}$

Add parentheses.

$\sqrt[3]{{(4 y^{2})}^{3} \cdot (4 y^{2})} \sqrt[3]{625 y^{5}}$

Step 2

Pull terms out from under the radical.

$4 y^{2} \sqrt[3]{4 y^{2}} \sqrt[3]{625 y^{5}}$

Step 3

Rewrite $625 y^{5}$ as ${(5 y)}^{3} \cdot (5 y^{2})$ .

Factor $125$ out of $625$ .

$4 y^{2} \sqrt[3]{4 y^{2}} \sqrt[3]{125 (5) y^{5}}$

Rewrite $125$ as $5^{3}$ .

$4 y^{2} \sqrt[3]{4 y^{2}} \sqrt[3]{5^{3} \cdot 5 y^{5}}$

Factor out $y^{3}$ .

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

Move $5$ .

$4 y^{2} \sqrt[3]{4 y^{2}} \sqrt[3]{5^{3} y^{3} \cdot 5 y^{2}}$

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

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

Add parentheses.

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

Step 4

Pull terms out from under the radical.

$4 y^{2} \sqrt[3]{4 y^{2}} (5 y \sqrt[3]{5 y^{2}})$

Step 5

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

Move $y$ .

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

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

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

$4 (y^{1} y^{2}) \sqrt[3]{4 y^{2}} (5 \sqrt[3]{5 y^{2}})$

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

$4 y^{1 + 2} \sqrt[3]{4 y^{2}} (5 \sqrt[3]{5 y^{2}})$

Add $1$ and $2$ .

$4 y^{3} \sqrt[3]{4 y^{2}} (5 \sqrt[3]{5 y^{2}})$

Step 6

Multiply $4 y^{3} \sqrt[3]{4 y^{2}} (5 \sqrt[3]{5 y^{2}})$ .

Multiply $5$ by $4$ .

$20 y^{3} \sqrt[3]{4 y^{2}} \sqrt[3]{5 y^{2}}$

Combine using the product rule for radicals.

$20 y^{3} \sqrt[3]{5 y^{2} (4 y^{2})}$

Multiply $4$ by $5$ .

$20 y^{3} \sqrt[3]{20 y^{2} y^{2}}$

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

Move $y^{2}$ .

$20 y^{3} \sqrt[3]{20 (y^{2} y^{2})}$

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

$20 y^{3} \sqrt[3]{20 y^{2 + 2}}$

Add $2$ and $2$ .

$20 y^{3} \sqrt[3]{20 y^{4}}$

Step 7

Rewrite $20 y^{4}$ as $y^{3} \cdot (20 y)$ .

Factor out $y^{3}$ .

$20 y^{3} \sqrt[3]{20 (y^{3} y)}$

Reorder $20$ and $y^{3}$ .

$20 y^{3} \sqrt[3]{y^{3} \cdot 20 y}$

Add parentheses.

$20 y^{3} \sqrt[3]{y^{3} \cdot (20 y)}$

Step 8

Pull terms out from under the radical.

$20 y^{3} (y \sqrt[3]{20 y})$

Step 9

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

Move $y$ .

$20 (y \cdot y^{3}) \sqrt[3]{20 y}$

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

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

$20 (y^{1} y^{3}) \sqrt[3]{20 y}$

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

$20 y^{1 + 3} \sqrt[3]{20 y}$

Add $1$ and $3$ .

$20 y^{4} \sqrt[3]{20 y}$

Simplify ( cube root of 256y^8)( cube root of 625y^5)

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