Solve for x (1/81)^(2x+9)=27^(6x-7)

${(\frac{1}{81})}^{2 x + 9} = 27^{6 x - 7}$

Step 1

Apply the product rule to $\frac{1}{81}$ .

$\frac{1^{2 x + 9}}{81^{2 x + 9}} = 27^{6 x - 7}$

Step 2

One to any power is one.

$\frac{1}{81^{2 x + 9}} = 27^{6 x - 7}$

Step 3

Move $81^{2 x + 9}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$81^{- (2 x + 9)} = 27^{6 x - 7}$

Step 4

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

$3^{4 (- (2 x + 9))} = 3^{3 (6 x - 7)}$

Step 5

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

$4 (- (2 x + 9)) = 3 (6 x - 7)$

Step 6

Solve for $x$ .

Simplify $4 (- (2 x + 9))$ .

Start simplifying.

$0 + 0 + 4 (- (2 x + 9)) = 3 (6 x - 7)$

Simplify by adding zeros.

$4 (- (2 x + 9)) = 3 (6 x - 7)$

Apply the distributive property.

$4 (- (2 x) - 1 \cdot 9) = 3 (6 x - 7)$

Multiply.

Multiply $2$ by $- 1$ .

$4 (- 2 x - 1 \cdot 9) = 3 (6 x - 7)$

Multiply $- 1$ by $9$ .

$4 (- 2 x - 9) = 3 (6 x - 7)$

Apply the distributive property.

$4 (- 2 x) + 4 \cdot - 9 = 3 (6 x - 7)$

Multiply.

Multiply $- 2$ by $4$ .

$- 8 x + 4 \cdot - 9 = 3 (6 x - 7)$

Multiply $4$ by $- 9$ .

$- 8 x - 36 = 3 (6 x - 7)$

Simplify $3 (6 x - 7)$ .

Apply the distributive property.

$- 8 x - 36 = 3 (6 x) + 3 \cdot - 7$

Multiply.

Multiply $6$ by $3$ .

$- 8 x - 36 = 18 x + 3 \cdot - 7$

Multiply $3$ by $- 7$ .

$- 8 x - 36 = 18 x - 21$

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

Subtract $18 x$ from both sides of the equation.

$- 8 x - 36 - 18 x = - 21$

Subtract $18 x$ from $- 8 x$ .

$- 26 x - 36 = - 21$

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

Add $36$ to both sides of the equation.

$- 26 x = - 21 + 36$

Add $- 21$ and $36$ .

$- 26 x = 15$

Divide each term in $- 26 x = 15$ by $- 26$ and simplify.

Divide each term in $- 26 x = 15$ by $- 26$ .

$\frac{- 26 x}{- 26} = \frac{15}{- 26}$

Simplify the left side.

Cancel the common factor of $- 26$ .

Cancel the common factor.

$\frac{- 26 x}{- 26} = \frac{15}{- 26}$

Divide $x$ by $1$ .

$x = \frac{15}{- 26}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{15}{26}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{15}{26}$

Decimal Form:

$x = - 0.5\overline{769230}$

Solve for x (1/81)^(2x+9)=27^(6x-7)

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