Solve the Rational Equation for x (5^(x/3))(5^(x/4))=5^5

$(5^{\frac{x}{3}}) (5^{\frac{x}{4}}) = 5^{5}$

Multiply $5^{\frac{x}{3}}$ by $5^{\frac{x}{4}}$ by adding the exponents.

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

$5^{\frac{x}{3} + \frac{x}{4}} = 5^{5}$

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

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

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

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

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

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

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

Multiply $3$ by $4$ .

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

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

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

Multiply $4$ by $3$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Move $4$ to the left of $x$ .

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

Move $3$ to the left of $x$ .

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

Add $4 x$ and $3 x$ .

$5^{\frac{7 x}{12}} = 5^{5}$

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

$\frac{7 x}{12} = 5$

Solve for $x$ .

Multiply both sides of the equation by $\frac{12}{7}$ .

$\frac{12}{7} \cdot \frac{7 x}{12} = \frac{12}{7} \cdot 5$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{12}{7} \cdot \frac{7 x}{12}$ .

Cancel the common factor of $12$ .

Cancel the common factor.

$\frac{12}{7} \cdot \frac{7 x}{12} = \frac{12}{7} \cdot 5$

Rewrite the expression.

$\frac{1}{7} (7 x) = \frac{12}{7} \cdot 5$

Cancel the common factor of $7$ .

Factor $7$ out of $7 x$ .

$\frac{1}{7} (7 (x)) = \frac{12}{7} \cdot 5$

Cancel the common factor.

$\frac{1}{7} (7 x) = \frac{12}{7} \cdot 5$

Rewrite the expression.

$x = \frac{12}{7} \cdot 5$

Simplify the right side.

Multiply $\frac{12}{7} \cdot 5$ .

Combine $\frac{12}{7}$ and $5$ .

$x = \frac{12 \cdot 5}{7}$

Multiply $12$ by $5$ .

$x = \frac{60}{7}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{60}{7}$

Decimal Form:

$x = 8.\overline{571428}$

Mixed Number Form:

$x = 8 \frac{4}{7}$

Solve the Rational Equation for x (5^(x/3))(5^(x/4))=5^5

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