Precalculus Examples

${(\frac{1}{2})}^{x + 14} = 4^{3 x}$

Step 1

Apply the product rule to $\frac{1}{2}$ .

$\frac{1^{x + 14}}{2^{x + 14}} = 4^{3 x}$

Step 2

One to any power is one.

$\frac{1}{2^{x + 14}} = 4^{3 x}$

Step 3

Move $2^{x + 14}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$2^{- (x + 14)} = 4^{3 x}$

Step 4

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

$2^{- (x + 14)} = 2^{2 (3 x)}$

Step 5

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

$- (x + 14) = 2 (3 x)$

Step 6

Solve for $x$ .

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Step 6.1

Simplify $- (x + 14)$ .

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Step 6.1.1

Rewrite.

$0 + 0 - (x + 14) = 2 \cdot (3 x)$

Step 6.1.2

Simplify by adding zeros.

$- (x + 14) = 2 \cdot (3 x)$

Step 6.1.3

Apply the distributive property.

$- x - 1 \cdot 14 = 2 \cdot (3 x)$

Step 6.1.4

Multiply $- 1$ by $14$ .

$- x - 14 = 2 \cdot (3 x)$

Step 6.2

Multiply $3$ by $2$ .

$- x - 14 = 6 x$

Step 6.3

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

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Step 6.3.1

Subtract $6 x$ from both sides of the equation.

$- x - 14 - 6 x = 0$

Step 6.3.2

Subtract $6 x$ from $- x$ .

$- 7 x - 14 = 0$

Step 6.4

Add $14$ to both sides of the equation.

$- 7 x = 14$

Step 6.5

Divide each term in $- 7 x = 14$ by $- 7$ and simplify.

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Step 6.5.1

Divide each term in $- 7 x = 14$ by $- 7$ .

$\frac{- 7 x}{- 7} = \frac{14}{- 7}$

Step 6.5.2

Simplify the left side.

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Step 6.5.2.1

Cancel the common factor of $- 7$ .

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Step 6.5.2.1.1

Cancel the common factor.

$\frac{- 7 x}{- 7} = \frac{14}{- 7}$

Step 6.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{14}{- 7}$

Step 6.5.3

Simplify the right side.

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Step 6.5.3.1

Divide $14$ by $- 7$ .

$x = - 2$