Algebra Examples

$\frac{\sqrt[5]{x^{4}}}{{(x^{\frac{1}{4}})}^{5}} = x^{a}$

Step 1

Rewrite the equation as $x^{a} = \frac{\sqrt[5]{x^{4}}}{{(x^{\frac{1}{4}})}^{5}}$ .

$x^{a} = \frac{\sqrt[5]{x^{4}}}{{(x^{\frac{1}{4}})}^{5}}$

Step 2

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

$x^{a} = \frac{x^{\frac{4}{5}}}{{(x^{\frac{1}{4}})}^{5}}$

Step 3

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

$x^{a} = \frac{x^{\frac{4}{5}}}{x^{\frac{1}{4} \cdot 5}}$

Step 4

Move $x^{\frac{1}{4} \cdot 5}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$x^{a} = x^{\frac{4}{5}} x^{- \frac{1}{4} \cdot 5}$

Step 5

Multiply $x^{\frac{4}{5}}$ by $x^{- \frac{1}{4} \cdot 5}$ by adding the exponents.

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Step 5.1

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

$x^{a} = x^{\frac{4}{5} - \frac{1}{4} \cdot 5}$

Step 5.2

Simplify each term.

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Step 5.2.1

Multiply $- \frac{1}{4} \cdot 5$ .

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Step 5.2.1.1

Multiply $5$ by $- 1$ .

$x^{a} = x^{\frac{4}{5} - 5 (\frac{1}{4})}$

Step 5.2.1.2

Combine $- 5$ and $\frac{1}{4}$ .

$x^{a} = x^{\frac{4}{5} + \frac{- 5}{4}}$

Step 5.2.2

Move the negative in front of the fraction.

$x^{a} = x^{\frac{4}{5} - \frac{5}{4}}$

Step 5.3

To write $\frac{4}{5}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x^{a} = x^{\frac{4}{5} \cdot \frac{4}{4} - \frac{5}{4}}$

Step 5.4

To write $- \frac{5}{4}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x^{a} = x^{\frac{4}{5} \cdot \frac{4}{4} - \frac{5}{4} \cdot \frac{5}{5}}$

Step 5.5

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.5.1

Multiply $\frac{4}{5}$ by $\frac{4}{4}$ .

$x^{a} = x^{\frac{4 \cdot 4}{5 \cdot 4} - \frac{5}{4} \cdot \frac{5}{5}}$

Step 5.5.2

Multiply $5$ by $4$ .

$x^{a} = x^{\frac{4 \cdot 4}{20} - \frac{5}{4} \cdot \frac{5}{5}}$

Step 5.5.3

Multiply $\frac{5}{4}$ by $\frac{5}{5}$ .

$x^{a} = x^{\frac{4 \cdot 4}{20} - \frac{5 \cdot 5}{4 \cdot 5}}$

Step 5.5.4

Multiply $4$ by $5$ .

$x^{a} = x^{\frac{4 \cdot 4}{20} - \frac{5 \cdot 5}{20}}$

Step 5.6

Combine the numerators over the common denominator.

$x^{a} = x^{\frac{4 \cdot 4 - 5 \cdot 5}{20}}$

Step 5.7

Simplify the numerator.

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Step 5.7.1

Multiply $4$ by $4$ .

$x^{a} = x^{\frac{16 - 5 \cdot 5}{20}}$

Step 5.7.2

Multiply $- 5$ by $5$ .

$x^{a} = x^{\frac{16 - 25}{20}}$

Step 5.7.3

Subtract $25$ from $16$ .

$x^{a} = x^{\frac{- 9}{20}}$

Step 5.8

Move the negative in front of the fraction.

$x^{a} = x^{- \frac{9}{20}}$

Step 6

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$a = - \frac{9}{20}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$a = - \frac{9}{20}$

Decimal Form:

$a = - 0.45$