Solve for x 3^(3x-7)=81^(12-3x)

$3^{3 x - 7} = 81^{12 - 3 x}$

Step 1

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

$3^{3 x - 7} = 3^{4 (12 - 3 x)}$

Step 2

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

$3 x - 7 = 4 (12 - 3 x)$

Step 3

Solve for $x$ .

Simplify $4 (12 - 3 x)$ .

Rewrite.

$3 x - 7 = 0 + 0 + 4 (12 - 3 x)$

Simplify by adding zeros.

$3 x - 7 = 4 (12 - 3 x)$

Apply the distributive property.

$3 x - 7 = 4 \cdot 12 + 4 (- 3 x)$

Multiply.

Multiply $4$ by $12$ .

$3 x - 7 = 48 + 4 (- 3 x)$

Multiply $- 3$ by $4$ .

$3 x - 7 = 48 - 12 x$

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

Add $12 x$ to both sides of the equation.

$3 x - 7 + 12 x = 48$

Add $3 x$ and $12 x$ .

$15 x - 7 = 48$

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

Add $7$ to both sides of the equation.

$15 x = 48 + 7$

Add $48$ and $7$ .

$15 x = 55$

Divide each term in $15 x = 55$ by $15$ and simplify.

Divide each term in $15 x = 55$ by $15$ .

$\frac{15 x}{15} = \frac{55}{15}$

Simplify the left side.

Cancel the common factor of $15$ .

Cancel the common factor.

$\frac{15 x}{15} = \frac{55}{15}$

Divide $x$ by $1$ .

$x = \frac{55}{15}$

Simplify the right side.

Cancel the common factor of $55$ and $15$ .

Factor $5$ out of $55$ .

$x = \frac{5 (11)}{15}$

Cancel the common factors.

Factor $5$ out of $15$ .

$x = \frac{5 \cdot 11}{5 \cdot 3}$

Cancel the common factor.

$x = \frac{5 \cdot 11}{5 \cdot 3}$

Rewrite the expression.

$x = \frac{11}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{11}{3}$

Decimal Form:

$x = 3.\overline{6}$

Mixed Number Form:

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

Solve for x 3^(3x-7)=81^(12-3x)

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