Trigonometry Examples

$9^{x - 2} = 27$

Step 1

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

$3^{2 (x - 2)} = 3^{3}$

Step 2

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

$2 (x - 2) = 3$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

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

$\frac{2 (x - 2)}{2} = \frac{3}{2}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $2$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{2 (x - 2)}{2} = \frac{3}{2}$

Step 3.1.2.1.2

Divide $x - 2$ by $1$ .

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

Step 3.2

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

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Step 3.2.1

Add $2$ to both sides of the equation.

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

Step 3.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.2.3

Combine $2$ and $\frac{2}{2}$ .

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

Step 3.2.4

Combine the numerators over the common denominator.

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

Step 3.2.5

Simplify the numerator.

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Step 3.2.5.1

Multiply $2$ by $2$ .

$x = \frac{3 + 4}{2}$

Step 3.2.5.2

Add $3$ and $4$ .

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 3.5$

Mixed Number Form:

$x = 3 \frac{1}{2}$