Solve for x (1/27)^(x+4)=3^4*3^(x-9)

${(\frac{1}{27})}^{x + 4} = 3^{4} \cdot 3^{x - 9}$

Step 1

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

$\frac{1^{x + 4}}{27^{x + 4}} = 3^{4} \cdot 3^{x - 9}$

Step 2

One to any power is one.

$\frac{1}{27^{x + 4}} = 3^{4} \cdot 3^{x - 9}$

Step 3

Move $27^{x + 4}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$27^{- (x + 4)} = 3^{4} \cdot 3^{x - 9}$

Step 4

Multiply $3^{4}$ by $3^{x - 9}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$27^{- (x + 4)} = 3^{4 + x - 9}$

Subtract $9$ from $4$ .

$27^{- (x + 4)} = 3^{x - 5}$

Step 5

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

$3^{3 (- (x + 4))} = 3^{x - 5}$

Step 6

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

$3 (- (x + 4)) = x - 5$

Step 7

Solve for $x$ .

Simplify $3 (- (x + 4))$ .

Rewrite.

$0 + 0 + 3 (- (x + 4)) = x - 5$

Simplify by adding zeros.

$3 (- (x + 4)) = x - 5$

Apply the distributive property.

$3 (- x - 1 \cdot 4) = x - 5$

Multiply $- 1$ by $4$ .

$3 (- x - 4) = x - 5$

Apply the distributive property.

$3 (- x) + 3 \cdot - 4 = x - 5$

Multiply.

Multiply $- 1$ by $3$ .

$- 3 x + 3 \cdot - 4 = x - 5$

Multiply $3$ by $- 4$ .

$- 3 x - 12 = x - 5$

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

Subtract $x$ from both sides of the equation.

$- 3 x - 12 - x = - 5$

Subtract $x$ from $- 3 x$ .

$- 4 x - 12 = - 5$

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

Add $12$ to both sides of the equation.

$- 4 x = - 5 + 12$

Add $- 5$ and $12$ .

$- 4 x = 7$

Divide each term in $- 4 x = 7$ by $- 4$ and simplify.

Divide each term in $- 4 x = 7$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{7}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{7}{- 4}$

Divide $x$ by $1$ .

$x = \frac{7}{- 4}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{7}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{7}{4}$

Decimal Form:

$x = - 1.75$

Mixed Number Form:

$x = - 1 \frac{3}{4}$

Solve for x (1/27)^(x+4)=3^4*3^(x-9)

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