Algebra Examples

$\frac{8^{x + 1} + 8^{x + 2}}{9} = 16$

Step 1

Multiply both sides of the equation by $9$ .

$9 \frac{8^{x + 1} + 8^{x + 2}}{9} = 9 \cdot 16$

Step 2

Simplify both sides of the equation.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Cancel the common factor of $9$ .

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Step 2.1.1.1

Cancel the common factor.

$9 \frac{8^{x + 1} + 8^{x + 2}}{9} = 9 \cdot 16$

Step 2.1.1.2

Rewrite the expression.

$8^{x + 1} + 8^{x + 2} = 9 \cdot 16$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply $9$ by $16$ .

$8^{x + 1} + 8^{x + 2} = 144$

Step 3

Factor out $8^{x + 1}$ from the expression.

$8^{x + 1} (1 + 8)$

Step 4

Add $1$ and $8$ .

$8^{x + 1} \cdot 9 = 144$

Step 5

Move $9$ to the left of $8^{x + 1}$ .

$9 \cdot 8^{x + 1} = 144$

Step 6

Divide each term in $9 \cdot 8^{x + 1} = 144$ by $9$ and simplify.

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Step 6.1

Divide each term in $9 \cdot 8^{x + 1} = 144$ by $9$ .

$\frac{9 \cdot 8^{x + 1}}{9} = \frac{144}{9}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $9$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{9 \cdot 8^{x + 1}}{9} = \frac{144}{9}$

Step 6.2.1.2

Divide $8^{x + 1}$ by $1$ .

$8^{x + 1} = \frac{144}{9}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $144$ by $9$ .

$8^{x + 1} = 16$

Step 7

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

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

Step 8

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

$3 (x + 1) = 4$

Step 9

Solve for $x$ .

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Step 9.1

Divide each term in $3 (x + 1) = 4$ by $3$ and simplify.

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Step 9.1.1

Divide each term in $3 (x + 1) = 4$ by $3$ .

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

Step 9.1.2

Simplify the left side.

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Step 9.1.2.1

Cancel the common factor of $3$ .

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Step 9.1.2.1.1

Cancel the common factor.

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

Step 9.1.2.1.2

Divide $x + 1$ by $1$ .

$x + 1 = \frac{4}{3}$

Step 9.2

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

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Step 9.2.1

Subtract $1$ from both sides of the equation.

$x = \frac{4}{3} - 1$

Step 9.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = \frac{4}{3} - 1 \cdot \frac{3}{3}$

Step 9.2.3

Combine $- 1$ and $\frac{3}{3}$ .

$x = \frac{4}{3} + \frac{- 1 \cdot 3}{3}$

Step 9.2.4

Combine the numerators over the common denominator.

$x = \frac{4 - 1 \cdot 3}{3}$

Step 9.2.5

Simplify the numerator.

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Step 9.2.5.1

Multiply $- 1$ by $3$ .

$x = \frac{4 - 3}{3}$

Step 9.2.5.2

Subtract $3$ from $4$ .

$x = \frac{1}{3}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{3}$

Decimal Form:

$x = 0.\overline{3}$