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.

Tap for more steps...

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 $1$ .

Tap for more steps...

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 $x$ .

Tap for more steps...

Step 4.1

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$x \cdot 12 = y \cdot 7$

Step 4.2

Divide each term in $x \cdot 12 = y \cdot 7$ by $12$ and simplify.

Tap for more steps...

Step 4.2.1

Divide each term in $x \cdot 12 = y \cdot 7$ by $12$ .

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

Step 4.2.2

Simplify the left side.

Tap for more steps...

Step 4.2.2.1

Cancel the common factor of $12$ .

Tap for more steps...

Step 4.2.2.1.1

Cancel the common factor.

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

Step 4.2.2.1.2

Divide $x$ by $1$ .

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

Step 4.2.3

Simplify the right side.

Tap for more steps...

Step 4.2.3.1

Move $7$ to the left of $y$ .

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

Step 5

To rewrite as a function of $y$ , write the equation so that $x$ is by itself on one side of the equal sign and an expression involving only $y$ is on the other side.

$f (y) = \frac{7 y}{12}$