Precalculus Examples

$4^{x + 2} = 16^{3 x - 1}$

Step 1

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

$4^{x + 2} = 4^{2 (3 x - 1)}$

Step 2

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

$x + 2 = 2 (3 x - 1)$

Step 3

Solve for $x$ .

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Step 3.1

Simplify $2 (3 x - 1)$ .

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Step 3.1.1

Rewrite.

$x + 2 = 0 + 0 + 2 (3 x - 1)$

Step 3.1.2

Simplify by adding zeros.

$x + 2 = 2 (3 x - 1)$

Step 3.1.3

Apply the distributive property.

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

Step 3.1.4

Multiply.

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Step 3.1.4.1

Multiply $3$ by $2$ .

$x + 2 = 6 x + 2 \cdot - 1$

Step 3.1.4.2

Multiply $2$ by $- 1$ .

$x + 2 = 6 x - 2$

Step 3.2

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

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Step 3.2.1

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

$x + 2 - 6 x = - 2$

Step 3.2.2

Subtract $6 x$ from $x$ .

$- 5 x + 2 = - 2$

Step 3.3

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

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Step 3.3.1

Subtract $2$ from both sides of the equation.

$- 5 x = - 2 - 2$

Step 3.3.2

Subtract $2$ from $- 2$ .

$- 5 x = - 4$

Step 3.4

Divide each term in $- 5 x = - 4$ by $- 5$ and simplify.

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Step 3.4.1

Divide each term in $- 5 x = - 4$ by $- 5$ .

$\frac{- 5 x}{- 5} = \frac{- 4}{- 5}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $- 5$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{- 5 x}{- 5} = \frac{- 4}{- 5}$

Step 3.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4}{- 5}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Dividing two negative values results in a positive value.

$x = \frac{4}{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4}{5}$

Decimal Form:

$x = 0.8$