Algebra Examples

$\frac{x^{\frac{1}{5}}}{\sqrt[5]{x^{6}}} = x^{a}$

Step 1

Rewrite the equation as $x^{a} = \frac{x^{\frac{1}{5}}}{\sqrt[5]{x^{6}}}$ .

$x^{a} = \frac{x^{\frac{1}{5}}}{\sqrt[5]{x^{6}}}$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{6}}$ as $x^{\frac{6}{5}}$ .

$x^{a} = \frac{x^{\frac{1}{5}}}{x^{\frac{6}{5}}}$

Step 3

Move $x^{\frac{6}{5}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$x^{a} = x^{\frac{1}{5}} x^{- \frac{6}{5}}$

Step 4

Multiply $x^{\frac{1}{5}}$ by $x^{- \frac{6}{5}}$ by adding the exponents.

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Step 4.1

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

$x^{a} = x^{\frac{1}{5} - \frac{6}{5}}$

Step 4.2

Combine the numerators over the common denominator.

$x^{a} = x^{\frac{1 - 6}{5}}$

Step 4.3

Subtract $6$ from $1$ .

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

Step 4.4

Divide $- 5$ by $5$ .

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

Step 5

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

$a = - 1$