Algebra Examples

$8 \sqrt[4]{5} - \sqrt[4]{20} \cdot \sqrt[4]{4}$

Step 1

Simplify each term.

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Step 1.1

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

$8 \sqrt[4]{5} - \sqrt[4]{20} \cdot \sqrt[4]{2^{2}}$

Step 1.2

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

$8 \sqrt[4]{5} - \sqrt[4]{20} \cdot \sqrt{\sqrt{2^{2}}}$

Step 1.3

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

$8 \sqrt[4]{5} - \sqrt[4]{20} \cdot \sqrt{2}$

Step 1.4

Multiply $- \sqrt[4]{20} \sqrt{2}$ .

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Step 1.4.1

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

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Step 1.4.1.1

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

$8 \sqrt[4]{5} - (2^{\frac{1}{2}} \sqrt[4]{20})$

Step 1.4.1.2

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

$8 \sqrt[4]{5} - (2^{\frac{2}{4}} \sqrt[4]{20})$

Step 1.4.1.3

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

$8 \sqrt[4]{5} - (\sqrt[4]{2^{2}} \sqrt[4]{20})$

Step 1.4.2

Combine using the product rule for radicals.

$8 \sqrt[4]{5} - \sqrt[4]{2^{2} \cdot 20}$

Step 1.4.3

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

$8 \sqrt[4]{5} - \sqrt[4]{4 \cdot 20}$

Step 1.4.4

Multiply $4$ by $20$ .

$8 \sqrt[4]{5} - \sqrt[4]{80}$

Step 1.5

Rewrite $80$ as $2^{4} \cdot 5$ .

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Step 1.5.1

Factor $16$ out of $80$ .

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

Step 1.5.2

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

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

Step 1.6

Pull terms out from under the radical.

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

Step 1.7

Multiply $2$ by $- 1$ .

$8 \sqrt[4]{5} - 2 \sqrt[4]{5}$

Step 2

Subtract $2 \sqrt[4]{5}$ from $8 \sqrt[4]{5}$ .

$6 \sqrt[4]{5}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$6 \sqrt[4]{5}$

Decimal Form:

$8.97209268 \dots$