Precalculus Examples

$2^{- 2 x + 8} = {(\frac{1}{8})}^{x - 3}$

Step 1

Apply the product rule to $\frac{1}{8}$ .

$2^{- 2 x + 8} = \frac{1^{x - 3}}{8^{x - 3}}$

Step 2

One to any power is one.

$2^{- 2 x + 8} = \frac{1}{8^{x - 3}}$

Step 3

Move $8^{x - 3}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$2^{- 2 x + 8} = 8^{- (x - 3)}$

Step 4

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

$2^{- 2 x + 8} = 2^{3 (- (x - 3))}$

Step 5

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

$- 2 x + 8 = 3 (- (x - 3))$

Step 6

Solve for $x$ .

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Step 6.1

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

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Step 6.1.1

Rewrite.

$- 2 x + 8 = 0 + 0 + 3 (- (x - 3))$

Step 6.1.2

Simplify by adding zeros.

$- 2 x + 8 = 3 (- (x - 3))$

Step 6.1.3

Apply the distributive property.

$- 2 x + 8 = 3 (- x - - 3)$

Step 6.1.4

Multiply $- 1$ by $- 3$ .

$- 2 x + 8 = 3 (- x + 3)$

Step 6.1.5

Apply the distributive property.

$- 2 x + 8 = 3 (- x) + 3 \cdot 3$

Step 6.1.6

Multiply.

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Step 6.1.6.1

Multiply $- 1$ by $3$ .

$- 2 x + 8 = - 3 x + 3 \cdot 3$

Step 6.1.6.2

Multiply $3$ by $3$ .

$- 2 x + 8 = - 3 x + 9$

Step 6.2

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

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Step 6.2.1

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

$- 2 x + 8 + 3 x = 9$

Step 6.2.2

Add $- 2 x$ and $3 x$ .

$x + 8 = 9$

Step 6.3

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

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Step 6.3.1

Subtract $8$ from both sides of the equation.

$x = 9 - 8$

Step 6.3.2

Subtract $8$ from $9$ .

$x = 1$