Precalculus Examples

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

Step 1

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

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

Step 2

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

$3 (6 + 3 x) = 2$

Step 3

Solve for $x$ .

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Step 3.1

Divide each term in $3 (6 + 3 x) = 2$ by $3$ and simplify.

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Step 3.1.1

Divide each term in $3 (6 + 3 x) = 2$ by $3$ .

$\frac{3 (6 + 3 x)}{3} = \frac{2}{3}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $3$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{3 (6 + 3 x)}{3} = \frac{2}{3}$

Step 3.1.2.1.2

Divide $6 + 3 x$ by $1$ .

$6 + 3 x = \frac{2}{3}$

Step 3.2

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

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Step 3.2.1

Subtract $6$ from both sides of the equation.

$3 x = \frac{2}{3} - 6$

Step 3.2.2

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$3 x = \frac{2}{3} - 6 \cdot \frac{3}{3}$

Step 3.2.3

Combine $- 6$ and $\frac{3}{3}$ .

$3 x = \frac{2}{3} + \frac{- 6 \cdot 3}{3}$

Step 3.2.4

Combine the numerators over the common denominator.

$3 x = \frac{2 - 6 \cdot 3}{3}$

Step 3.2.5

Simplify the numerator.

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Step 3.2.5.1

Multiply $- 6$ by $3$ .

$3 x = \frac{2 - 18}{3}$

Step 3.2.5.2

Subtract $18$ from $2$ .

$3 x = \frac{- 16}{3}$

Step 3.2.6

Move the negative in front of the fraction.

$3 x = - \frac{16}{3}$

Step 3.3

Divide each term in $3 x = - \frac{16}{3}$ by $3$ and simplify.

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Step 3.3.1

Divide each term in $3 x = - \frac{16}{3}$ by $3$ .

$\frac{3 x}{3} = \frac{- \frac{16}{3}}{3}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $3$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- \frac{16}{3}}{3}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{16}{3}}{3}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{16}{3} \cdot \frac{1}{3}$

Step 3.3.3.2

Multiply $- \frac{16}{3} \cdot \frac{1}{3}$ .

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Step 3.3.3.2.1

Multiply $\frac{1}{3}$ by $\frac{16}{3}$ .

$x = - \frac{16}{3 \cdot 3}$

Step 3.3.3.2.2

Multiply $3$ by $3$ .

$x = - \frac{16}{9}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{16}{9}$

Decimal Form:

$x = - 1.\overline{7}$

Mixed Number Form:

$x = - 1 \frac{7}{9}$