Solve for x ((5^x)^(1/3))/(5^(2/3))=5

$\frac{{(5^{x})}^{\frac{1}{3}}}{5^{\frac{2}{3}}} = 5$

Step 1

Multiply both sides of the equation by $5^{\frac{2}{3}}$ .

$5^{\frac{2}{3}} \frac{{(5^{x})}^{\frac{1}{3}}}{5^{\frac{2}{3}}} = 5^{\frac{2}{3}} \cdot 5$

Step 2

Simplify both sides of the equation.

Simplify the left side.

Simplify $5^{\frac{2}{3}} \frac{{(5^{x})}^{\frac{1}{3}}}{5^{\frac{2}{3}}}$ .

Cancel the common factor of $5^{\frac{2}{3}}$ .

Cancel the common factor.

$5^{\frac{2}{3}} \frac{{(5^{x})}^{\frac{1}{3}}}{5^{\frac{2}{3}}} = 5^{\frac{2}{3}} \cdot 5$

Rewrite the expression.

${(5^{x})}^{\frac{1}{3}} = 5^{\frac{2}{3}} \cdot 5$

Multiply the exponents in ${(5^{x})}^{\frac{1}{3}}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$5^{x \frac{1}{3}} = 5^{\frac{2}{3}} \cdot 5$

Combine $x$ and $\frac{1}{3}$ .

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

Simplify the right side.

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

Multiply $5^{\frac{2}{3}}$ by $5$ .

Raise $5$ to the power of $1$ .

$5^{\frac{x}{3}} = 5^{\frac{2}{3}} \cdot 5^{1}$

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

$5^{\frac{x}{3}} = 5^{\frac{2}{3} + 1}$

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Add $2$ and $3$ .

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

Step 3

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

$\frac{x}{3} = \frac{5}{3}$

Step 4

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$x = 5$