Solve for x 3^(7x+4)=(1/27)^(x-3)

$3^{7 x + 4} = {(\frac{1}{27})}^{x - 3}$

Step 1

Apply the product rule to $\frac{1}{27}$ .

$3^{7 x + 4} = \frac{1^{x - 3}}{27^{x - 3}}$

Step 2

One to any power is one.

$3^{7 x + 4} = \frac{1}{27^{x - 3}}$

Step 3

Move $27^{x - 3}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$3^{7 x + 4} = 27^{- (x - 3)}$

Step 4

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

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

Step 5

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

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

Step 6

Solve for $x$ .

Simplify $3 (- (x - 3))$ .

Rewrite.

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

Simplify by adding zeros.

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

Apply the distributive property.

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

Multiply $- 1$ by $- 3$ .

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

Apply the distributive property.

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

Multiply.

Multiply $- 1$ by $3$ .

$7 x + 4 = - 3 x + 3 \cdot 3$

Multiply $3$ by $3$ .

$7 x + 4 = - 3 x + 9$

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

Add $3 x$ to both sides of the equation.

$7 x + 4 + 3 x = 9$

Add $7 x$ and $3 x$ .

$10 x + 4 = 9$

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

Subtract $4$ from both sides of the equation.

$10 x = 9 - 4$

Subtract $4$ from $9$ .

$10 x = 5$

Divide each term in $10 x = 5$ by $10$ and simplify.

Divide each term in $10 x = 5$ by $10$ .

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

Simplify the left side.

Cancel the common factor of $10$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Cancel the common factor of $5$ and $10$ .

Factor $5$ out of $5$ .

$x = \frac{5 (1)}{10}$

Cancel the common factors.

Factor $5$ out of $10$ .

$x = \frac{5 \cdot 1}{5 \cdot 2}$

Cancel the common factor.

$x = \frac{5 \cdot 1}{5 \cdot 2}$

Rewrite the expression.

$x = \frac{1}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{2}$

Decimal Form:

$x = 0.5$

Solve for x 3^(7x+4)=(1/27)^(x-3)

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