Solve for x fifth root of 32^x=1/8

$\sqrt[5]{32^{x}} = \frac{1}{8}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{32^{x}}$ as $32^{\frac{x}{5}}$ .

$32^{\frac{x}{5}} = \frac{1}{8}$

Step 2

Raise $8$ to the power of $1$ .

$32^{\frac{x}{5}} = \frac{1}{8^{1}}$

Step 3

Move $8^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$32^{\frac{x}{5}} = 8^{- 1}$

Step 4

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

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

Step 5

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

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

Step 6

Solve for $x$ .

Combine $5$ and $\frac{x}{5}$ .

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

Divide each term in $\frac{5 x}{5} = - 3$ by $5$ and simplify.

Divide each term in $\frac{5 x}{5} = - 3$ by $5$ .

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

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

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

$x = - 3$