Algebra Examples

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

Step 1

Raise $2$ to the power of $3$ .

$\sqrt[3]{2} \cdot \sqrt[5]{8}$

Step 2

Multiply $\sqrt[3]{2} \sqrt[5]{8}$ .

Tap for more steps...

Step 2.1

Rewrite the expression using the least common index of $15$ .

Tap for more steps...

Step 2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2}$ as $2^{\frac{1}{3}}$ .

$2^{\frac{1}{3}} \sqrt[5]{8}$

Step 2.1.2

Rewrite $2^{\frac{1}{3}}$ as $2^{\frac{5}{15}}$ .

$2^{\frac{5}{15}} \sqrt[5]{8}$

Step 2.1.3

Rewrite $2^{\frac{5}{15}}$ as $\sqrt[15]{2^{5}}$ .

$\sqrt[15]{2^{5}} \sqrt[5]{8}$

Step 2.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{8}$ as $8^{\frac{1}{5}}$ .

$\sqrt[15]{2^{5}} \cdot 8^{\frac{1}{5}}$

Step 2.1.5

Rewrite $8^{\frac{1}{5}}$ as $8^{\frac{3}{15}}$ .

$\sqrt[15]{2^{5}} \cdot 8^{\frac{3}{15}}$

Step 2.1.6

Rewrite $8^{\frac{3}{15}}$ as $\sqrt[15]{8^{3}}$ .

$\sqrt[15]{2^{5}} \sqrt[15]{8^{3}}$

Step 2.2

Combine using the product rule for radicals.

$\sqrt[15]{2^{5} \cdot 8^{3}}$

Step 2.3

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

$\sqrt[15]{2^{5} \cdot {(2^{3})}^{3}}$

Step 2.4

Multiply the exponents in ${(2^{3})}^{3}$ .

Tap for more steps...

Step 2.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt[15]{2^{5} \cdot 2^{3 \cdot 3}}$

Step 2.4.2

Multiply $3$ by $3$ .

$\sqrt[15]{2^{5} \cdot 2^{9}}$

Step 2.5

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

$\sqrt[15]{2^{5 + 9}}$

Step 2.6

Add $5$ and $9$ .

$\sqrt[15]{2^{14}}$

Step 3

Raise $2$ to the power of $14$ .

$\sqrt[15]{16384}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\sqrt[15]{16384}$

Decimal Form:

$1.90968320 \dots$