Trigonometry Examples

$3^{x} - 3^{x - 2} = 8$

Step 1

Factor out $3^{x}$ from the expression.

$3^{x} (1 - \frac{1}{9})$

Step 2

Simplify the expression.

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Step 2.1

Write $1$ as a fraction with a common denominator.

$3^{x} (\frac{9}{9} - \frac{1}{9}) = 8$

Step 2.2

Combine the numerators over the common denominator.

$3^{x} (\frac{9 - 1}{9}) = 8$

Step 2.3

Subtract $1$ from $9$ .

$3^{x} (\frac{8}{9}) = 8$

Step 3

Combine $3^{x}$ and $\frac{8}{9}$ .

$\frac{3^{x} \cdot 8}{9} = 8$

Step 4

Move $8$ to the left of $3^{x}$ .

$\frac{8 \cdot 3^{x}}{9} = 8$

Step 5

Multiply both sides of the equation by $\frac{9}{8}$ .

$\frac{9}{8} \cdot \frac{8 \cdot 3^{x}}{9} = \frac{9}{8} \cdot 8$

Step 6

Simplify both sides of the equation.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Simplify $\frac{9}{8} \cdot \frac{8 \cdot 3^{x}}{9}$ .

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Step 6.1.1.1

Combine.

$\frac{9 (8 \cdot 3^{x})}{8 \cdot 9} = \frac{9}{8} \cdot 8$

Step 6.1.1.2

Cancel the common factor of $9$ .

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Step 6.1.1.2.1

Cancel the common factor.

$\frac{9 (8 \cdot 3^{x})}{8 \cdot 9} = \frac{9}{8} \cdot 8$

Step 6.1.1.2.2

Rewrite the expression.

$\frac{8 \cdot 3^{x}}{8} = \frac{9}{8} \cdot 8$

Step 6.1.1.3

Cancel the common factor of $8$ .

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Step 6.1.1.3.1

Cancel the common factor.

$\frac{8 \cdot 3^{x}}{8} = \frac{9}{8} \cdot 8$

Step 6.1.1.3.2

Divide $3^{x}$ by $1$ .

$3^{x} = \frac{9}{8} \cdot 8$

Step 6.2

Simplify the right side.

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Step 6.2.1

Cancel the common factor of $8$ .

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Step 6.2.1.1

Cancel the common factor.

$3^{x} = \frac{9}{8} \cdot 8$

Step 6.2.1.2

Rewrite the expression.

$3^{x} = 9$

Step 7

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

$3^{x} = 3^{2}$

Step 8

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

$x = 2$