Solve for x (1/64)^(x/2+1)=(1/16)^(x/3-3)

${(\frac{1}{64})}^{\frac{x}{2} + 1} = {(\frac{1}{16})}^{\frac{x}{3} - 3}$

Step 1

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

${(\frac{1}{4})}^{3 (\frac{x}{2} + 1)} = {(\frac{1}{4})}^{2 (\frac{x}{3} - 3)}$

Step 2

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

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

Step 3

Solve for $x$ .

Simplify $3 (\frac{x}{2} + 1)$ .

Start simplifying.

$0 + 0 + 3 (\frac{x}{2} + 1) = 2 (\frac{x}{3} - 3)$

Simplify by adding zeros.

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

Apply the distributive property.

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

Combine $3$ and $\frac{x}{2}$ .

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

Multiply $3$ by $1$ .

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

Simplify $2 (\frac{x}{3} - 3)$ .

Apply the distributive property.

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

Combine $2$ and $\frac{x}{3}$ .

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

Multiply $2$ by $- 3$ .

$\frac{3 x}{2} + 3 = \frac{2 x}{3} - 6$

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

Subtract $\frac{2 x}{3}$ from both sides of the equation.

$\frac{3 x}{2} + 3 - \frac{2 x}{3} = - 6$

To write $\frac{3 x}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{3 x}{2} \cdot \frac{3}{3} - \frac{2 x}{3} + 3 = - 6$

To write $- \frac{2 x}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 x}{2} \cdot \frac{3}{3} - \frac{2 x}{3} \cdot \frac{2}{2} + 3 = - 6$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3 x}{2}$ by $\frac{3}{3}$ .

$\frac{3 x \cdot 3}{2 \cdot 3} - \frac{2 x}{3} \cdot \frac{2}{2} + 3 = - 6$

Multiply $2$ by $3$ .

$\frac{3 x \cdot 3}{6} - \frac{2 x}{3} \cdot \frac{2}{2} + 3 = - 6$

Multiply $\frac{2 x}{3}$ by $\frac{2}{2}$ .

$\frac{3 x \cdot 3}{6} - \frac{2 x \cdot 2}{3 \cdot 2} + 3 = - 6$

Multiply $3$ by $2$ .

$\frac{3 x \cdot 3}{6} - \frac{2 x \cdot 2}{6} + 3 = - 6$

Combine the numerators over the common denominator.

$\frac{3 x \cdot 3 - 2 x \cdot 2}{6} + 3 = - 6$

Simplify each term.

Simplify the numerator.

Factor $x$ out of $3 x \cdot 3 - 2 x \cdot 2$ .

Factor $x$ out of $3 x \cdot 3$ .

$\frac{x (3 \cdot 3) - 2 x \cdot 2}{6} + 3 = - 6$

Factor $x$ out of $- 2 x \cdot 2$ .

$\frac{x (3 \cdot 3) + x (- 2 \cdot 2)}{6} + 3 = - 6$

Factor $x$ out of $x (3 \cdot 3) + x (- 2 \cdot 2)$ .

$\frac{x (3 \cdot 3 - 2 \cdot 2)}{6} + 3 = - 6$

Multiply $3$ by $3$ .

$\frac{x (9 - 2 \cdot 2)}{6} + 3 = - 6$

Multiply $- 2$ by $2$ .

$\frac{x (9 - 4)}{6} + 3 = - 6$

Subtract $4$ from $9$ .

$\frac{x \cdot 5}{6} + 3 = - 6$

Move $5$ to the left of $x$ .

$\frac{5 x}{6} + 3 = - 6$

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

Subtract $3$ from both sides of the equation.

$\frac{5 x}{6} = - 6 - 3$

Subtract $3$ from $- 6$ .

$\frac{5 x}{6} = - 9$

Multiply both sides of the equation by $\frac{6}{5}$ .

$\frac{6}{5} \cdot \frac{5 x}{6} = \frac{6}{5} \cdot - 9$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{6}{5} \cdot \frac{5 x}{6}$ .

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6}{5} \cdot \frac{5 x}{6} = \frac{6}{5} \cdot - 9$

Rewrite the expression.

$\frac{1}{5} (5 x) = \frac{6}{5} \cdot - 9$

Cancel the common factor of $5$ .

Factor $5$ out of $5 x$ .

$\frac{1}{5} (5 (x)) = \frac{6}{5} \cdot - 9$

Cancel the common factor.

$\frac{1}{5} (5 x) = \frac{6}{5} \cdot - 9$

Rewrite the expression.

$x = \frac{6}{5} \cdot - 9$

Simplify the right side.

Simplify $\frac{6}{5} \cdot - 9$ .

Multiply $\frac{6}{5} \cdot - 9$ .

Combine $\frac{6}{5}$ and $- 9$ .

$x = \frac{6 \cdot - 9}{5}$

Multiply $6$ by $- 9$ .

$x = \frac{- 54}{5}$

Move the negative in front of the fraction.

$x = - \frac{54}{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{54}{5}$

Decimal Form:

$x = - 10.8$

Mixed Number Form:

$x = - 10 \frac{4}{5}$

Solve for x (1/64)^(x/2+1)=(1/16)^(x/3-3)

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