Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Multiply $x^{5 (\frac{1}{4})}$ by $x^{- \frac{1}{4}}$ 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.

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

Step 5.2

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

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

Step 5.3

Combine the numerators over the common denominator.

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

Step 5.4

Subtract $1$ from $5$ .

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

Step 5.5

Divide $4$ by $4$ .

$x^{a} = x^{1}$

Step 6

Simplify $x^{1}$ .

$x^{a} = x$

Step 7

Create equivalent expressions in the equation that all have equal bases.

$x^{a} = x^{1}$

Step 8

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

$a = 1$