Solve for k (1/216)^(-2k)=(1/36)^(k+2)

${(\frac{1}{216})}^{- 2 k} = {(\frac{1}{36})}^{k + 2}$

Step 1

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

${(\frac{1}{6})}^{3 (- 2 k)} = {(\frac{1}{6})}^{2 (k + 2)}$

Step 2

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

$3 (- 2 k) = 2 (k + 2)$

Step 3

Solve for $k$ .

Multiply $- 2$ by $3$ .

$- 6 k = 2 (k + 2)$

Simplify $2 (k + 2)$ .

Apply the distributive property.

$- 6 k = 2 k + 2 \cdot 2$

Multiply $2$ by $2$ .

$- 6 k = 2 k + 4$

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

Subtract $2 k$ from both sides of the equation.

$- 6 k - 2 k = 4$

Subtract $2 k$ from $- 6 k$ .

$- 8 k = 4$

Divide each term in $- 8 k = 4$ by $- 8$ and simplify.

Divide each term in $- 8 k = 4$ by $- 8$ .

$\frac{- 8 k}{- 8} = \frac{4}{- 8}$

Simplify the left side.

Cancel the common factor of $- 8$ .

Cancel the common factor.

$\frac{- 8 k}{- 8} = \frac{4}{- 8}$

Divide $k$ by $1$ .

$k = \frac{4}{- 8}$

Simplify the right side.

Cancel the common factor of $4$ and $- 8$ .

Factor $4$ out of $4$ .

$k = \frac{4 (1)}{- 8}$

Cancel the common factors.

Factor $4$ out of $- 8$ .

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

Cancel the common factor.

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

Rewrite the expression.

$k = \frac{1}{- 2}$

Move the negative in front of the fraction.

$k = - \frac{1}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$k = - \frac{1}{2}$

Decimal Form:

$k = - 0.5$

Solve for k (1/216)^(-2k)=(1/36)^(k+2)

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