Solve for x 27=(1/3)^(x/2)

$27 = {(\frac{1}{3})}^{\frac{x}{2}}$

Step 1

Rewrite the equation as ${(\frac{1}{3})}^{\frac{x}{2}} = 27$ .

${(\frac{1}{3})}^{\frac{x}{2}} = 27$

Step 2

Apply the product rule to $\frac{1}{3}$ .

$\frac{1^{\frac{x}{2}}}{3^{\frac{x}{2}}} = 27$

Step 3

One to any power is one.

$\frac{1}{3^{\frac{x}{2}}} = 27$

Step 4

Move $3^{\frac{x}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$3^{- \frac{x}{2}} = 27$

Step 5

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

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

Step 6

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

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

Step 7

Solve for $x$ .

Multiply both sides of the equation by $- 2$ .

$- 2 (- \frac{x}{2}) = - 2 \cdot 3$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- 2 (- \frac{x}{2})$ .

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{x}{2}$ into the numerator.

$- 2 \frac{- x}{2} = - 2 \cdot 3$

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{- x}{2} = - 2 \cdot 3$

Cancel the common factor.

$2 \cdot - 1 \frac{- x}{2} = - 2 \cdot 3$

Rewrite the expression.

$- - x = - 2 \cdot 3$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 x = - 2 \cdot 3$

Multiply $x$ by $1$ .

$x = - 2 \cdot 3$

Simplify the right side.

Multiply $- 2$ by $3$ .

$x = - 6$