Simplify ( cube root of 625z)( cube root of 40z^6)

$(\sqrt[3]{625 z}) (\sqrt[3]{40 z^{6}})$

Step 1

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

Factor $125$ out of $625$ .

$\sqrt[3]{125 (5) z} \sqrt[3]{40 z^{6}}$

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

$\sqrt[3]{5^{3} \cdot 5 z} \sqrt[3]{40 z^{6}}$

Add parentheses.

$\sqrt[3]{5^{3} \cdot (5 z)} \sqrt[3]{40 z^{6}}$

Step 2

Pull terms out from under the radical.

$5 \sqrt[3]{5 z} \sqrt[3]{40 z^{6}}$

Step 3

Rewrite $40 z^{6}$ as ${(2 z^{2})}^{3} \cdot 5$ .

Factor $8$ out of $40$ .

$5 \sqrt[3]{5 z} \sqrt[3]{8 (5) z^{6}}$

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

$5 \sqrt[3]{5 z} \sqrt[3]{2^{3} \cdot 5 z^{6}}$

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

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

Move $5$ .

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

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

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

Multiply $2$ by $5$ .

$10 \sqrt[3]{5 z} (z^{2} \sqrt[3]{5})$

Combine using the product rule for radicals.

$10 (z^{2} \sqrt[3]{5 z \cdot 5})$

Multiply $5$ by $5$ .

$10 (z^{2} \sqrt[3]{25 z})$

$10 z^{2} \sqrt[3]{25 z}$