Trigonometry Examples

$25^{2 x + 1} = 125$

Step 1

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

$5^{2 (2 x + 1)} = 5^{3}$

Step 2

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

$2 (2 x + 1) = 3$

Step 3

Solve for $x$ .

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Step 3.1

Divide each term in $2 (2 x + 1) = 3$ by $2$ and simplify.

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Step 3.1.1

Divide each term in $2 (2 x + 1) = 3$ by $2$ .

$\frac{2 (2 x + 1)}{2} = \frac{3}{2}$

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.

$\frac{2 (2 x + 1)}{2} = \frac{3}{2}$

Step 3.1.2.1.2

Divide $2 x + 1$ by $1$ .

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

Step 3.2

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

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Step 3.2.1

Subtract $1$ from both sides of the equation.

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

Step 3.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.2.3

Combine $- 1$ and $\frac{2}{2}$ .

$2 x = \frac{3}{2} + \frac{- 1 \cdot 2}{2}$

Step 3.2.4

Combine the numerators over the common denominator.

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

Step 3.2.5

Simplify the numerator.

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Step 3.2.5.1

Multiply $- 1$ by $2$ .

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

Step 3.2.5.2

Subtract $2$ from $3$ .

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

Step 3.3

Divide each term in $2 x = \frac{1}{2}$ by $2$ and simplify.

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Step 3.3.1

Divide each term in $2 x = \frac{1}{2}$ by $2$ .

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $2$ .

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Step 3.3.2.1.1

Cancel the common factor.

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

Step 3.3.2.1.2

Divide $x$ by $1$ .

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

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.

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

Step 3.3.3.2

Multiply $\frac{1}{2} \cdot \frac{1}{2}$ .

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Step 3.3.3.2.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

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

Step 3.3.3.2.2

Multiply $2$ by $2$ .

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.25$