Solve the Rational Equation for x 6^(1/3)*6^(1/4)=6^(x/y)

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

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

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

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

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

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

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

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

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

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

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

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

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

Multiply $3$ by $4$ .

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

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

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

Multiply $4$ by $3$ .

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

Combine the numerators over the common denominator.

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

Add $4$ and $3$ .

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

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

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

Solve for $x$ .

Multiply both sides by $y$ .

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

Simplify.

Simplify the left side.

Cancel the common factor of $y$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the right side.

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

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