Solve for x fifth root of square root of 125=5^x

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

Step 1

Rewrite the equation as $5^{x} = \sqrt[5]{\sqrt{125}}$ .

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

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{125}$ as $125^{\frac{1}{2}}$ .

$5^{x} = \sqrt[5]{125^{\frac{1}{2}}}$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{125^{\frac{1}{2}}}$ as $125^{\frac{\frac{1}{2}}{5}}$ .

$5^{x} = 125^{\frac{\frac{1}{2}}{5}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$5^{x} = 125^{\frac{1}{2} \cdot \frac{1}{5}}$

Step 5

Multiply $\frac{1}{2} \cdot \frac{1}{5}$ .

Multiply $\frac{1}{2}$ by $\frac{1}{5}$ .

$5^{x} = 125^{\frac{1}{2 \cdot 5}}$

Multiply $2$ by $5$ .

$5^{x} = 125^{\frac{1}{10}}$

Step 6

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

$5^{x} = 5^{3 (\frac{1}{10})}$

Step 7

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

$x = 3 (\frac{1}{10})$

Step 8

Combine $3$ and $\frac{1}{10}$ .

$x = \frac{3}{10}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3}{10}$

Decimal Form:

$x = 0.3$