Solve for x (1/27)^(x-6)=27

${(\frac{1}{27})}^{x - 6} = 27$

Step 1

Apply the product rule to $\frac{1}{27}$ .

$\frac{1^{x - 6}}{27^{x - 6}} = 27$

Step 2

One to any power is one.

$\frac{1}{27^{x - 6}} = 27$

Step 3

Move $27^{x - 6}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$27^{- (x - 6)} = 27$

Step 4

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

$27^{- (x - 6)} = 27^{1}$

Step 5

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

$- (x - 6) = 1$

Step 6

Solve for $x$ .

Simplify $- (x - 6)$ .

Apply the distributive property.

$- x - - 6 = 1$

Multiply $- 1$ by $- 6$ .

$- x + 6 = 1$

Move all terms not containing $x$ to the right side of the equation.

Subtract $6$ from both sides of the equation.

$- x = 1 - 6$

Subtract $6$ from $1$ .

$- x = - 5$

Divide each term in $- x = - 5$ by $- 1$ and simplify.

Divide each term in $- x = - 5$ by $- 1$ .

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $- 5$ by $- 1$ .

$x = 5$