Solve for x (5^(x/6))(5^((2x)/3))=5^4

$(5^{\frac{x}{6}}) (5^{\frac{2 x}{3}}) = 5^{4}$

Multiply $5^{\frac{x}{6}}$ by $5^{\frac{2 x}{3}}$ by adding the exponents.

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

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

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

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

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

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

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

Multiply $3$ by $2$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Factor $x$ out of $x + 2 x \cdot 2$ .

Raise $x$ to the power of $1$ .

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

Factor $x$ out of $x^{1}$ .

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

Factor $x$ out of $2 x \cdot 2$ .

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

Factor $x$ out of $x \cdot 1 + x (2 \cdot 2)$ .

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

Multiply $2$ by $2$ .

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

Add $1$ and $4$ .

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

Move $5$ to the left of $x$ .

$5^{\frac{5 x}{6}} = 5^{4}$

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

$\frac{5 x}{6} = 4$

Solve for $x$ .

Multiply both sides of the equation by $\frac{6}{5}$ .

$\frac{6}{5} \cdot \frac{5 x}{6} = \frac{6}{5} \cdot 4$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{6}{5} \cdot \frac{5 x}{6}$ .

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6}{5} \cdot \frac{5 x}{6} = \frac{6}{5} \cdot 4$

Rewrite the expression.

$\frac{1}{5} (5 x) = \frac{6}{5} \cdot 4$

Cancel the common factor of $5$ .

Factor $5$ out of $5 x$ .

$\frac{1}{5} (5 (x)) = \frac{6}{5} \cdot 4$

Cancel the common factor.

$\frac{1}{5} (5 x) = \frac{6}{5} \cdot 4$

Rewrite the expression.

$x = \frac{6}{5} \cdot 4$

Simplify the right side.

Multiply $\frac{6}{5} \cdot 4$ .

Combine $\frac{6}{5}$ and $4$ .

$x = \frac{6 \cdot 4}{5}$

Multiply $6$ by $4$ .

$x = \frac{24}{5}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{24}{5}$

Decimal Form:

$x = 4.8$

Mixed Number Form:

$x = 4 \frac{4}{5}$

Solve for x (5^(x/6))(5^((2x)/3))=5^4

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