Algebra Examples

$x^{\frac{1}{6}} \sqrt[3]{x^{2}} = x^{a}$

Step 1

Rewrite the equation as $x^{a} = x^{\frac{1}{6}} \sqrt[3]{x^{2}}$ .

$x^{a} = x^{\frac{1}{6}} \sqrt[3]{x^{2}}$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

$x^{a} = x^{\frac{1}{6}} x^{\frac{2}{3}}$

Step 3

Multiply $x^{\frac{1}{6}}$ by $x^{\frac{2}{3}}$ by adding the exponents.

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Step 3.1

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

$x^{a} = x^{\frac{1}{6} + \frac{2}{3}}$

Step 3.2

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

$x^{a} = x^{\frac{1}{6} + \frac{2}{3} \cdot \frac{2}{2}}$

Step 3.3

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

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Step 3.3.1

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

$x^{a} = x^{\frac{1}{6} + \frac{2 \cdot 2}{3 \cdot 2}}$

Step 3.3.2

Multiply $3$ by $2$ .

$x^{a} = x^{\frac{1}{6} + \frac{2 \cdot 2}{6}}$

Step 3.4

Combine the numerators over the common denominator.

$x^{a} = x^{\frac{1 + 2 \cdot 2}{6}}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $2$ by $2$ .

$x^{a} = x^{\frac{1 + 4}{6}}$

Step 3.5.2

Add $1$ and $4$ .

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

Step 4

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

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$a = 0.8\overline{3}$