Algebra Examples

$5 \sqrt[4]{64} \cdot 2 \sqrt[4]{8}$

Step 1

Rewrite $64$ as $2^{4} \cdot 4$ .

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Step 1.1

Factor $16$ out of $64$ .

$5 \sqrt[4]{16 (4)} \cdot (2 \sqrt[4]{8})$

Step 1.2

Rewrite $16$ as $2^{4}$ .

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

Step 2

Pull terms out from under the radical.

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

Step 3

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

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

Step 4

Rewrite $\sqrt[4]{2^{2}}$ as $\sqrt{\sqrt{2^{2}}}$ .

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

Step 5

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

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

Step 6

Multiply $2$ by $5$ .

$10 \sqrt{2} \cdot (2 \sqrt[4]{8})$

Step 7

Multiply $10 \sqrt{2} (2 \sqrt[4]{8})$ .

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Step 7.1

Multiply $2$ by $10$ .

$20 \sqrt{2} \sqrt[4]{8}$

Step 7.2

Rewrite the expression using the least common index of $4$ .

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Step 7.2.1

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

$20 (\sqrt[4]{8} \cdot 2^{\frac{1}{2}})$

Step 7.2.2

Rewrite $2^{\frac{1}{2}}$ as $2^{\frac{2}{4}}$ .

$20 (\sqrt[4]{8} \cdot 2^{\frac{2}{4}})$

Step 7.2.3

Rewrite $2^{\frac{2}{4}}$ as $\sqrt[4]{2^{2}}$ .

$20 (\sqrt[4]{8} \sqrt[4]{2^{2}})$

Step 7.3

Combine using the product rule for radicals.

$20 \sqrt[4]{8 \cdot 2^{2}}$

Step 7.4

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

$20 \sqrt[4]{2^{3} \cdot 2^{2}}$

Step 7.5

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

$20 \sqrt[4]{2^{3 + 2}}$

Step 7.6

Add $3$ and $2$ .

$20 \sqrt[4]{2^{5}}$

Step 8

Raise $2$ to the power of $5$ .

$20 \sqrt[4]{32}$

Step 9

Rewrite $32$ as $2^{4} \cdot 2$ .

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Step 9.1

Factor $16$ out of $32$ .

$20 \sqrt[4]{16 (2)}$

Step 9.2

Rewrite $16$ as $2^{4}$ .

$20 \sqrt[4]{2^{4} \cdot 2}$

Step 10

Pull terms out from under the radical.

$20 (2 \sqrt[4]{2})$

Step 11

Multiply $2$ by $20$ .

$40 \sqrt[4]{2}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$40 \sqrt[4]{2}$

Decimal Form:

$47.56828460 \dots$