Solve for x 3125^x=1/( square root of 5)

$3125^{x} = \frac{1}{\sqrt{5}}$

Step 1

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

$3125^{x} = \frac{1}{5^{\frac{1}{2}}}$

Step 2

Move $5^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$3125^{x} = 5^{- \frac{1}{2}}$

Step 3

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

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

Step 4

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

$5 x = - \frac{1}{2}$

Step 5

Divide each term in $5 x = - \frac{1}{2}$ by $5$ and simplify.

Divide each term in $5 x = - \frac{1}{2}$ by $5$ .

$\frac{5 x}{5} = \frac{- \frac{1}{2}}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{- \frac{1}{2}}{5}$

Divide $x$ by $1$ .

$x = \frac{- \frac{1}{2}}{5}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{1}{2} \cdot \frac{1}{5}$

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

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

$x = - \frac{1}{5 \cdot 2}$

Multiply $5$ by $2$ .

$x = - \frac{1}{10}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1}{10}$

Decimal Form:

$x = - 0.1$