Basic Math Examples

$\sqrt[3]{16} \cdot \sqrt[5]{4}$

Step 1

Rewrite $16$ as $2^{3} \cdot 2$ .

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Step 1.1

Factor $8$ out of $16$ .

$\sqrt[3]{8 (2)} \cdot \sqrt[5]{4}$

Step 1.2

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

$\sqrt[3]{2^{3} \cdot 2} \cdot \sqrt[5]{4}$

Step 2

Pull terms out from under the radical.

$2 \sqrt[3]{2} \cdot \sqrt[5]{4}$

Step 3

Multiply $2 \sqrt[3]{2} \sqrt[5]{4}$ .

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Step 3.1

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

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Step 3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{4}$ as $4^{\frac{1}{5}}$ .

$2 (4^{\frac{1}{5}} \sqrt[3]{2})$

Step 3.1.2

Rewrite $4^{\frac{1}{5}}$ as $4^{\frac{3}{15}}$ .

$2 (4^{\frac{3}{15}} \sqrt[3]{2})$

Step 3.1.3

Rewrite $4^{\frac{3}{15}}$ as $\sqrt[15]{4^{3}}$ .

$2 (\sqrt[15]{4^{3}} \sqrt[3]{2})$

Step 3.1.4

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

$2 (\sqrt[15]{4^{3}} \cdot 2^{\frac{1}{3}})$

Step 3.1.5

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

$2 (\sqrt[15]{4^{3}} \cdot 2^{\frac{5}{15}})$

Step 3.1.6

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

$2 (\sqrt[15]{4^{3}} \sqrt[15]{2^{5}})$

Step 3.2

Combine using the product rule for radicals.

$2 \sqrt[15]{4^{3} \cdot 2^{5}}$

Step 3.3

Rewrite $4$ as $2^{2}$ .

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

Multiply $2$ by $3$ .

$2 \sqrt[15]{2^{6} \cdot 2^{5}}$

Step 3.5

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

$2 \sqrt[15]{2^{6 + 5}}$

Step 3.6

Add $6$ and $5$ .

$2 \sqrt[15]{2^{11}}$

Step 4

Raise $2$ to the power of $11$ .

$2 \sqrt[15]{2048}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt[15]{2048}$

Decimal Form:

$3.32495158 \dots$