Trigonometry Examples

25^{x} = \frac{1}{125}

$25^{x} = \frac{1}{125}$

Step 1

Raise

125

$125$ to the power of

1

$1$ .

25^{x} = \frac{1}{125^{1}}

$25^{x} = \frac{1}{125^{1}}$

Step 2

Move

125^{1}

$125^{1}$ to the numerator using the negative exponent rule

\frac{1}{b^{n}} = b^{- n}

$\frac{1}{b^{n}} = b^{- n}$ .

25^{x} = 125^{- 1}

$25^{x} = 125^{- 1}$

Step 3

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

5^{2 x} = 5^{- 3}

$5^{2 x} = 5^{- 3}$

Step 4

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

2 x = - 3

$2 x = - 3$

Step 5

Divide each term in

2 x = - 3

$2 x = - 3$ by

2

$2$ and simplify.

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Step 5.1

Divide each term in

2 x = - 3

$2 x = - 3$ by

2

$2$ .

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

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

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of

2

$2$ .

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Step 5.2.1.1

Cancel the common factor.

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

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

Step 5.2.1.2

Divide

x

$x$ by

1

$1$ .

x = \frac{- 3}{2}

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

x = \frac{- 3}{2}

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

x = \frac{- 3}{2}

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

Step 5.3

Simplify the right side.

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Step 5.3.1

Move the negative in front of the fraction.

x = - \frac{3}{2}

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

x = - \frac{3}{2}

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

x = - \frac{3}{2}

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

Step 6

The result can be shown in multiple forms.

Exact Form:

x = - \frac{3}{2}

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

Decimal Form:

x = - 1.5

$x = - 1.5$

Mixed Number Form:

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

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

25^{x} = \frac{1}{125}

$25^{x} = \frac{1}{125}$

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