Solve for x 625^x=1/( square root of 5)

$625^{x} = \frac{1}{\sqrt{5}}$

Step 1

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

$625^{x} = \frac{1}{5^{\frac{1}{2}}}$

Step 2

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

$625^{x} = 5^{- \frac{1}{2}}$

Step 3

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

$5^{4 x} = 5^{- \frac{1}{2}}$

Step 4

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

$4 x = - \frac{1}{2}$

Step 5

Divide each term in $4 x = - \frac{1}{2}$ by $4$ and simplify.

Divide each term in $4 x = - \frac{1}{2}$ by $4$ .

$\frac{4 x}{4} = \frac{- \frac{1}{2}}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{- \frac{1}{2}}{4}$

Divide $x$ by $1$ .

$x = \frac{- \frac{1}{2}}{4}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{1}{2} \cdot \frac{1}{4}$

Multiply $- \frac{1}{2} \cdot \frac{1}{4}$ .

Multiply $\frac{1}{4}$ by $\frac{1}{2}$ .

$x = - \frac{1}{4 \cdot 2}$

Multiply $4$ by $2$ .

$x = - \frac{1}{8}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1}{8}$

Decimal Form:

$x = - 0.125$