Algebra Examples

$6^{\frac{1}{3}} \cdot 6^{\frac{1}{4}} = 6^{\frac{x}{y}}$

Step 1

Rewrite the equation as $6^{\frac{x}{y}} = 6^{\frac{1}{3}} \cdot 6^{\frac{1}{4}}$ .

$6^{\frac{x}{y}} = 6^{\frac{1}{3}} \cdot 6^{\frac{1}{4}}$

Step 2

Multiply $6^{\frac{1}{3}}$ by $6^{\frac{1}{4}}$ 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.

$6^{\frac{x}{y}} = 6^{\frac{1}{3} + \frac{1}{4}}$

Step 2.2

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

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

Step 2.3

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

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

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

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

Step 2.4.2

Multiply $3$ by $4$ .

$6^{\frac{x}{y}} = 6^{\frac{4}{12} + \frac{1}{4} \cdot \frac{3}{3}}$

Step 2.4.3

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

$6^{\frac{x}{y}} = 6^{\frac{4}{12} + \frac{3}{4 \cdot 3}}$

Step 2.4.4

Multiply $4$ by $3$ .

$6^{\frac{x}{y}} = 6^{\frac{4}{12} + \frac{3}{12}}$

Step 2.5

Combine the numerators over the common denominator.

$6^{\frac{x}{y}} = 6^{\frac{4 + 3}{12}}$

Step 2.6

Add $4$ and $3$ .

$6^{\frac{x}{y}} = 6^{\frac{7}{12}}$

Step 3

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

$\frac{x}{y} = \frac{7}{12}$

Step 4

Solve for $x$ .

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Step 4.1

Multiply both sides by $y$ .

$\frac{x}{y} y = \frac{7}{12} y$

Step 4.2

Simplify.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Cancel the common factor of $y$ .

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Step 4.2.1.1.1

Cancel the common factor.

$\frac{x}{y} y = \frac{7}{12} y$

Step 4.2.1.1.2

Rewrite the expression.

$x = \frac{7}{12} y$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Combine $\frac{7}{12}$ and $y$ .

$x = \frac{7 y}{12}$