Simplify ( cube root of 45z^5)( cube root of 4z)

$(\sqrt[3]{45 z^{5}}) (\sqrt[3]{4 z})$

Step 1

Rewrite $45 z^{5}$ as $z^{3} \cdot (45 z^{2})$ .

Factor out $z^{3}$ .

$\sqrt[3]{45 (z^{3} z^{2})} \sqrt[3]{4 z}$

Reorder $45$ and $z^{3}$ .

$\sqrt[3]{z^{3} \cdot 45 z^{2}} \sqrt[3]{4 z}$

Add parentheses.

$\sqrt[3]{z^{3} \cdot (45 z^{2})} \sqrt[3]{4 z}$

Step 2

Pull terms out from under the radical.

$z \sqrt[3]{45 z^{2}} \sqrt[3]{4 z}$

Step 3

Multiply $z \sqrt[3]{45 z^{2}} \sqrt[3]{4 z}$ .

Combine using the product rule for radicals.

$z \sqrt[3]{4 z (45 z^{2})}$

Multiply $45$ by $4$ .

$z \sqrt[3]{180 z \cdot z^{2}}$

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

Move $z^{2}$ .

$z \sqrt[3]{180 (z^{2} z)}$

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

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

$z \sqrt[3]{180 (z^{2} z^{1})}$

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

$z \sqrt[3]{180 z^{2 + 1}}$

Add $2$ and $1$ .

$z \sqrt[3]{180 z^{3}}$

Step 4

Reorder $180$ and $z^{3}$ .

$z \sqrt[3]{z^{3} \cdot 180}$

Step 5

Pull terms out from under the radical.

$z (z \sqrt[3]{180})$

Step 6

Rewrite using the commutative property of multiplication.

$\sqrt[3]{180} z \cdot z$

Step 7

Multiply $z$ by $z$ by adding the exponents.

Move $z$ .

$\sqrt[3]{180} (z \cdot z)$

Multiply $z$ by $z$ .

$\sqrt[3]{180} z^{2}$