Simplify cube root of z^13 ninth root of z^6

$\sqrt[3]{z^{13}} \sqrt[9]{z^{6}}$

Step 1

Rewrite $z^{13}$ as ${(z^{4})}^{3} z$ .

Factor out $z^{12}$ .

$\sqrt[3]{z^{12} z} \sqrt[9]{z^{6}}$

Rewrite $z^{12}$ as ${(z^{4})}^{3}$ .

$\sqrt[3]{{(z^{4})}^{3} z} \sqrt[9]{z^{6}}$

Step 2

Pull terms out from under the radical.

$z^{4} \sqrt[3]{z} \sqrt[9]{z^{6}}$

Step 3

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

$z^{4} \sqrt[3]{z} \sqrt[9]{{(z^{2})}^{3}}$

Step 4

Rewrite $\sqrt[9]{{(z^{2})}^{3}}$ as $\sqrt[3]{\sqrt[3]{{(z^{2})}^{3}}}$ .

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

Step 5

Pull terms out from under the radical, assuming real numbers.

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

Step 6

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

Combine using the product rule for radicals.

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

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

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

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

$z^{4} \sqrt[3]{z^{2} z^{1}}$

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

$z^{4} \sqrt[3]{z^{2 + 1}}$

Add $2$ and $1$ .

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

Step 7

Pull terms out from under the radical, assuming real numbers.

$z^{4} z$

Step 8

Multiply $z^{4}$ by $z$ by adding the exponents.

Multiply $z^{4}$ by $z$ .

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

$z^{4} z^{1}$

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

$z^{4 + 1}$

Add $4$ and $1$ .

$z^{5}$