Solve for x 32^(-3x)=(1/4)^(-3x)

$32^{- 3 x} = {(\frac{1}{4})}^{- 3 x}$

Step 1

Apply the product rule to $\frac{1}{4}$ .

$32^{- 3 x} = \frac{1^{- 3 x}}{4^{- 3 x}}$

Step 2

One to any power is one.

$32^{- 3 x} = \frac{1}{4^{- 3 x}}$

Step 3

Move $4^{- 3 x}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$32^{- 3 x} = 4^{3 x}$

Step 4

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

$2^{5 (- 3 x)} = 2^{2 (3 x)}$

Step 5

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

$5 (- 3 x) = 2 (3 x)$

Step 6

Solve for $x$ .

Multiply $- 3$ by $5$ .

$- 15 x = 2 \cdot (3 x)$

Multiply $3$ by $2$ .

$- 15 x = 6 x$

Move all terms containing $x$ to the left side of the equation.

Subtract $6 x$ from both sides of the equation.

$- 15 x - 6 x = 0$

Subtract $6 x$ from $- 15 x$ .

$- 21 x = 0$

Divide each term in $- 21 x = 0$ by $- 21$ and simplify.

Divide each term in $- 21 x = 0$ by $- 21$ .

$\frac{- 21 x}{- 21} = \frac{0}{- 21}$

Simplify the left side.

Cancel the common factor of $- 21$ .

Cancel the common factor.

$\frac{- 21 x}{- 21} = \frac{0}{- 21}$

Divide $x$ by $1$ .

$x = \frac{0}{- 21}$

Simplify the right side.

Divide $0$ by $- 21$ .

$x = 0$