Precalculus Examples

$8^{6 + 3 x} = 4^{x + 1}$

Step 1

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

$2^{3 (6 + 3 x)} = 2^{2 (x + 1)}$

Step 2

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

$3 (6 + 3 x) = 2 (x + 1)$

Step 3

Solve for $x$ .

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Step 3.1

Simplify $3 (6 + 3 x)$ .

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Step 3.1.1

Rewrite.

$0 + 0 + 3 (6 + 3 x) = 2 (x + 1)$

Step 3.1.2

Simplify by adding zeros.

$3 (6 + 3 x) = 2 (x + 1)$

Step 3.1.3

Apply the distributive property.

$3 \cdot 6 + 3 (3 x) = 2 (x + 1)$

Step 3.1.4

Multiply.

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Step 3.1.4.1

Multiply $3$ by $6$ .

$18 + 3 (3 x) = 2 (x + 1)$

Step 3.1.4.2

Multiply $3$ by $3$ .

$18 + 9 x = 2 (x + 1)$

Step 3.2

Simplify $2 (x + 1)$ .

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Step 3.2.1

Apply the distributive property.

$18 + 9 x = 2 x + 2 \cdot 1$

Step 3.2.2

Multiply $2$ by $1$ .

$18 + 9 x = 2 x + 2$

Step 3.3

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

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Step 3.3.1

Subtract $2 x$ from both sides of the equation.

$18 + 9 x - 2 x = 2$

Step 3.3.2

Subtract $2 x$ from $9 x$ .

$18 + 7 x = 2$

Step 3.4

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

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Step 3.4.1

Subtract $18$ from both sides of the equation.

$7 x = 2 - 18$

Step 3.4.2

Subtract $18$ from $2$ .

$7 x = - 16$

Step 3.5

Divide each term in $7 x = - 16$ by $7$ and simplify.

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Step 3.5.1

Divide each term in $7 x = - 16$ by $7$ .

$\frac{7 x}{7} = \frac{- 16}{7}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Cancel the common factor of $7$ .

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Step 3.5.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{- 16}{7}$

Step 3.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 16}{7}$

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Move the negative in front of the fraction.

$x = - \frac{16}{7}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{16}{7}$

Decimal Form:

$x = - 2.\overline{285714}$

Mixed Number Form:

$x = - 2 \frac{2}{7}$