Algebra Examples

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

Step 1

Rewrite $z^{5}$ as ${(z^{2})}^{2} z$ .

Tap for more steps...

Step 1.1

Factor out $z^{4}$ .

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

Step 1.2

Rewrite $z^{4}$ as ${(z^{2})}^{2}$ .

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

Step 2

Pull terms out from under the radical.

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

Step 3

Factor out $z^{3}$ .

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

Tap for more steps...

Step 5.1

Move $z$ .

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

Step 5.2

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

Tap for more steps...

Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.3

Add $1$ and $2$ .

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

Step 6

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

Tap for more steps...

Step 6.1

Rewrite the expression using the least common index of $6$ .

Tap for more steps...

Step 6.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{z^{2}}$ as $z^{\frac{2}{3}}$ .

$z^{3} (z^{\frac{2}{3}} \sqrt{z})$

Step 6.1.2

Rewrite $z^{\frac{2}{3}}$ as $z^{\frac{4}{6}}$ .

$z^{3} (z^{\frac{4}{6}} \sqrt{z})$

Step 6.1.3

Rewrite $z^{\frac{4}{6}}$ as $\sqrt[6]{z^{4}}$ .

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

Step 6.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{z}$ as $z^{\frac{1}{2}}$ .

$z^{3} (\sqrt[6]{z^{4}} z^{\frac{1}{2}})$

Step 6.1.5

Rewrite $z^{\frac{1}{2}}$ as $z^{\frac{3}{6}}$ .

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

Step 6.1.6

Rewrite $z^{\frac{3}{6}}$ as $\sqrt[6]{z^{3}}$ .

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

Step 6.2

Combine using the product rule for radicals.

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

Step 6.3

Multiply $z^{4}$ by $z^{3}$ by adding the exponents.

Tap for more steps...

Step 6.3.1

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

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

Step 6.3.2

Add $4$ and $3$ .

$z^{3} \sqrt[6]{z^{7}}$

Step 7

Factor out $z^{6}$ .

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

Step 8

Pull terms out from under the radical.

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

Step 9

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

Tap for more steps...

Step 9.1

Move $z$ .

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

Step 9.2

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

Tap for more steps...

Step 9.2.1

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

$z^{1} z^{3} \sqrt[6]{z}$

Step 9.2.2

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

$z^{1 + 3} \sqrt[6]{z}$

Step 9.3

Add $1$ and $3$ .

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