Algebra Examples

$\frac{{(x^{\frac{1}{6}})}^{5}}{\sqrt[3]{x^{5}}} = x^{a}$

Step 1

Rewrite the equation as $x^{a} = \frac{{(x^{\frac{1}{6}})}^{5}}{\sqrt[3]{x^{5}}}$ .

$x^{a} = \frac{{(x^{\frac{1}{6}})}^{5}}{\sqrt[3]{x^{5}}}$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{5}}$ as $x^{\frac{5}{3}}$ .

$x^{a} = \frac{{(x^{\frac{1}{6}})}^{5}}{x^{\frac{5}{3}}}$

Step 3

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

$x^{a} = \frac{x^{\frac{1}{6} \cdot 5}}{x^{\frac{5}{3}}}$

Step 4

Move $x^{\frac{5}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$x^{a} = x^{\frac{1}{6} \cdot 5} x^{- \frac{5}{3}}$

Step 5

Multiply $x^{\frac{1}{6} \cdot 5}$ by $x^{- \frac{5}{3}}$ 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.

$x^{a} = x^{\frac{1}{6} \cdot 5 - \frac{5}{3}}$

Step 5.2

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

$x^{a} = x^{\frac{5}{6} - \frac{5}{3}}$

Step 5.3

To write $- \frac{5}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x^{a} = x^{\frac{5}{6} - \frac{5}{3} \cdot \frac{2}{2}}$

Step 5.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.4.1

Multiply $\frac{5}{3}$ by $\frac{2}{2}$ .

$x^{a} = x^{\frac{5}{6} - \frac{5 \cdot 2}{3 \cdot 2}}$

Step 5.4.2

Multiply $3$ by $2$ .

$x^{a} = x^{\frac{5}{6} - \frac{5 \cdot 2}{6}}$

Step 5.5

Combine the numerators over the common denominator.

$x^{a} = x^{\frac{5 - 5 \cdot 2}{6}}$

Step 5.6

Simplify the numerator.

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Step 5.6.1

Multiply $- 5$ by $2$ .

$x^{a} = x^{\frac{5 - 10}{6}}$

Step 5.6.2

Subtract $10$ from $5$ .

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

Step 5.7

Move the negative in front of the fraction.

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

Step 6

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

$a = - \frac{5}{6}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$a = - \frac{5}{6}$

Decimal Form:

$a = - 0.8\overline{3}$