Solve for x ( fourth root of 5)^x=125

${(\sqrt[4]{5})}^{x} = 125$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{5}$ as $5^{\frac{1}{4}}$ .

${(5^{\frac{1}{4}})}^{x} = 125$

Step 2

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

$5^{\frac{1}{4} x} = 125$

Step 3

Create equivalent expressions in the equation that all have equal bases.

$5^{\frac{1}{4} x} = 5^{3}$

Step 4

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

$\frac{1}{4} x = 3$

Step 5

Solve for $x$ .

Multiply both sides of the equation by $4$ .

$4 (\frac{1}{4} x) = 4 \cdot 3$

Simplify both sides of the equation.

Simplify the left side.

Simplify $4 (\frac{1}{4} x)$ .

Combine $\frac{1}{4}$ and $x$ .

$4 \frac{x}{4} = 4 \cdot 3$

Cancel the common factor of $4$ .

Cancel the common factor.

$4 \frac{x}{4} = 4 \cdot 3$

Rewrite the expression.

$x = 4 \cdot 3$

Simplify the right side.

Multiply $4$ by $3$ .

$x = 12$