Algebra Examples

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

Step 1

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

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

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{x}$ as $x^{\frac{1}{6}}$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Multiply $x^{\frac{1}{6}}$ by $x^{- 5 (\frac{1}{6})}$ 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.

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

Step 5.2

Simplify each term.

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Step 5.2.1

Combine $- 5$ and $\frac{1}{6}$ .

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

Step 5.2.2

Move the negative in front of the fraction.

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

Step 5.3

Combine the numerators over the common denominator.

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

Step 5.4

Subtract $5$ from $1$ .

$x^{a} = x^{\frac{- 4}{6}}$

Step 5.5

Cancel the common factor of $- 4$ and $6$ .

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Step 5.5.1

Factor $2$ out of $- 4$ .

$x^{a} = x^{\frac{2 (- 2)}{6}}$

Step 5.5.2

Cancel the common factors.

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Step 5.5.2.1

Factor $2$ out of $6$ .

$x^{a} = x^{\frac{2 \cdot - 2}{2 \cdot 3}}$

Step 5.5.2.2

Cancel the common factor.

$x^{a} = x^{\frac{2 \cdot - 2}{2 \cdot 3}}$

Step 5.5.2.3

Rewrite the expression.

$x^{a} = x^{\frac{- 2}{3}}$

Step 5.6

Move the negative in front of the fraction.

$x^{a} = x^{- \frac{2}{3}}$

Step 6

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

$a = - \frac{2}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$a = - \frac{2}{3}$

Decimal Form:

$a = - 0.\overline{6}$