Simplify ( fourth root of 5-4)/( fourth root of 20)

$\frac{\sqrt[4]{5} - 4}{\sqrt[4]{20}}$

Step 1

Multiply $\frac{\sqrt[4]{5} - 4}{\sqrt[4]{20}}$ by $\frac{{\sqrt[4]{20}}^{3}}{{\sqrt[4]{20}}^{3}}$ .

$\frac{\sqrt[4]{5} - 4}{\sqrt[4]{20}} \cdot \frac{{\sqrt[4]{20}}^{3}}{{\sqrt[4]{20}}^{3}}$

Step 2

Combine and simplify the denominator.

Multiply $\frac{\sqrt[4]{5} - 4}{\sqrt[4]{20}}$ by $\frac{{\sqrt[4]{20}}^{3}}{{\sqrt[4]{20}}^{3}}$ .

$\frac{(\sqrt[4]{5} - 4) {\sqrt[4]{20}}^{3}}{\sqrt[4]{20} {\sqrt[4]{20}}^{3}}$

Raise $\sqrt[4]{20}$ to the power of $1$ .

$\frac{(\sqrt[4]{5} - 4) {\sqrt[4]{20}}^{3}}{{\sqrt[4]{20}}^{1} {\sqrt[4]{20}}^{3}}$

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

$\frac{(\sqrt[4]{5} - 4) {\sqrt[4]{20}}^{3}}{{\sqrt[4]{20}}^{1 + 3}}$

Add $1$ and $3$ .

$\frac{(\sqrt[4]{5} - 4) {\sqrt[4]{20}}^{3}}{{\sqrt[4]{20}}^{4}}$

Rewrite ${\sqrt[4]{20}}^{4}$ as $20$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{20}$ as $20^{\frac{1}{4}}$ .

$\frac{(\sqrt[4]{5} - 4) {\sqrt[4]{20}}^{3}}{{(20^{\frac{1}{4}})}^{4}}$

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

$\frac{(\sqrt[4]{5} - 4) {\sqrt[4]{20}}^{3}}{20^{\frac{1}{4} \cdot 4}}$

Combine $\frac{1}{4}$ and $4$ .

$\frac{(\sqrt[4]{5} - 4) {\sqrt[4]{20}}^{3}}{20^{\frac{4}{4}}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{(\sqrt[4]{5} - 4) {\sqrt[4]{20}}^{3}}{20^{\frac{4}{4}}}$

Rewrite the expression.

$\frac{(\sqrt[4]{5} - 4) {\sqrt[4]{20}}^{3}}{20^{1}}$

Evaluate the exponent.

$\frac{(\sqrt[4]{5} - 4) {\sqrt[4]{20}}^{3}}{20}$

Step 3

Simplify the numerator.

Rewrite ${\sqrt[4]{20}}^{3}$ as ${(20^{3})}^{\frac{1}{4}}$ .

$\frac{(\sqrt[4]{5} - 4) \sqrt[4]{20^{3}}}{20}$

Raise $20$ to the power of $3$ .

$\frac{(\sqrt[4]{5} - 4) \sqrt[4]{8000}}{20}$

Rewrite $8000$ as $2^{4} \cdot 500$ .

Factor $16$ out of $8000$ .

$\frac{(\sqrt[4]{5} - 4) \sqrt[4]{16 (500)}}{20}$

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

$\frac{(\sqrt[4]{5} - 4) \sqrt[4]{2^{4} \cdot 500}}{20}$

Pull terms out from under the radical.

$\frac{(\sqrt[4]{5} - 4) \cdot 2 \sqrt[4]{500}}{20}$

Step 4

Cancel the common factor of $2$ and $20$ .

Factor $2$ out of $(\sqrt[4]{5} - 4) \cdot 2 \sqrt[4]{500}$ .

$\frac{2 ((\sqrt[4]{5} - 4) \sqrt[4]{500})}{20}$

Cancel the common factors.

Factor $2$ out of $20$ .

$\frac{2 ((\sqrt[4]{5} - 4) \sqrt[4]{500})}{2 \cdot 10}$

Cancel the common factor.

$\frac{2 ((\sqrt[4]{5} - 4) \sqrt[4]{500})}{2 \cdot 10}$

Rewrite the expression.

$\frac{(\sqrt[4]{5} - 4) \sqrt[4]{500}}{10}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{(\sqrt[4]{5} - 4) \sqrt[4]{500}}{10}$

Decimal Form:

$- 1.18437643 \dots$

Simplify ( fourth root of 5-4)/( fourth root of 20)

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