Algebra Examples

$x^{\frac{1}{3}} \cdot x^{\frac{1}{2}} \cdot x^{\frac{1}{4}}$

Step 1

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

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Step 1.1

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

$x^{\frac{1}{3} + \frac{1}{2}} \cdot x^{\frac{1}{4}}$

Step 1.2

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x^{\frac{1}{3} \cdot \frac{2}{2} + \frac{1}{2}} \cdot x^{\frac{1}{4}}$

Step 1.3

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x^{\frac{1}{3} \cdot \frac{2}{2} + \frac{1}{2} \cdot \frac{3}{3}} \cdot x^{\frac{1}{4}}$

Step 1.4

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

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Step 1.4.1

Multiply $\frac{1}{3}$ by $\frac{2}{2}$ .

$x^{\frac{2}{3 \cdot 2} + \frac{1}{2} \cdot \frac{3}{3}} \cdot x^{\frac{1}{4}}$

Step 1.4.2

Multiply $3$ by $2$ .

$x^{\frac{2}{6} + \frac{1}{2} \cdot \frac{3}{3}} \cdot x^{\frac{1}{4}}$

Step 1.4.3

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

$x^{\frac{2}{6} + \frac{3}{2 \cdot 3}} \cdot x^{\frac{1}{4}}$

Step 1.4.4

Multiply $2$ by $3$ .

$x^{\frac{2}{6} + \frac{3}{6}} \cdot x^{\frac{1}{4}}$

Step 1.5

Combine the numerators over the common denominator.

$x^{\frac{2 + 3}{6}} \cdot x^{\frac{1}{4}}$

Step 1.6

Add $2$ and $3$ .

$x^{\frac{5}{6}} \cdot x^{\frac{1}{4}}$

Step 2

Multiply $x^{\frac{5}{6}}$ by $x^{\frac{1}{4}}$ by adding the exponents.

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Step 2.1

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

$x^{\frac{5}{6} + \frac{1}{4}}$

Step 2.2

To write $\frac{5}{6}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x^{\frac{5}{6} \cdot \frac{2}{2} + \frac{1}{4}}$

Step 2.3

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x^{\frac{5}{6} \cdot \frac{2}{2} + \frac{1}{4} \cdot \frac{3}{3}}$

Step 2.4

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{5}{6}$ by $\frac{2}{2}$ .

$x^{\frac{5 \cdot 2}{6 \cdot 2} + \frac{1}{4} \cdot \frac{3}{3}}$

Step 2.4.2

Multiply $6$ by $2$ .

$x^{\frac{5 \cdot 2}{12} + \frac{1}{4} \cdot \frac{3}{3}}$

Step 2.4.3

Multiply $\frac{1}{4}$ by $\frac{3}{3}$ .

$x^{\frac{5 \cdot 2}{12} + \frac{3}{4 \cdot 3}}$

Step 2.4.4

Multiply $4$ by $3$ .

$x^{\frac{5 \cdot 2}{12} + \frac{3}{12}}$

Step 2.5

Combine the numerators over the common denominator.

$x^{\frac{5 \cdot 2 + 3}{12}}$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $5$ by $2$ .

$x^{\frac{10 + 3}{12}}$

Step 2.6.2

Add $10$ and $3$ .

$x^{\frac{13}{12}}$

Step 3

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\sqrt[12]{x^{13}}$