Trigonometry Examples

$5^{2 - x} = \frac{1}{125}$

Step 1

Raise $125$ to the power of $1$ .

$5^{2 - x} = \frac{1}{125^{1}}$

Step 2

Move $125^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$5^{2 - x} = 125^{- 1}$

Step 3

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

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

Step 4

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

$2 - x = - 3$

Step 5

Solve for $x$ .

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Step 5.1

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

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Step 5.1.1

Subtract $2$ from both sides of the equation.

$- x = - 3 - 2$

Step 5.1.2

Subtract $2$ from $- 3$ .

$- x = - 5$

Step 5.2

Divide each term in $- x = - 5$ by $- 1$ and simplify.

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Step 5.2.1

Divide each term in $- x = - 5$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 5}{- 1}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 5}{- 1}$

Step 5.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 5}{- 1}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $- 5$ by $- 1$ .

$x = 5$