Precalculus Examples

${(\frac{1}{5})}^{2 - x} = 25$

Step 1

Apply the product rule to $\frac{1}{5}$ .

$\frac{1^{2 - x}}{5^{2 - x}} = 25$

Step 2

One to any power is one.

$\frac{1}{5^{2 - x}} = 25$

Step 3

Move $5^{2 - x}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$5^{- (2 - x)} = 25$

Step 4

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

$5^{- (2 - x)} = 5^{2}$

Step 5

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

$- (2 - x) = 2$

Step 6

Solve for $x$ .

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Step 6.1

Simplify $- (2 - x)$ .

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Step 6.1.1

Apply the distributive property.

$- 1 \cdot 2 - - x = 2$

Step 6.1.2

Multiply $- 1$ by $2$ .

$- 2 - - x = 2$

Step 6.1.3

Multiply $- - x$ .

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Step 6.1.3.1

Multiply $- 1$ by $- 1$ .

$- 2 + 1 x = 2$

Step 6.1.3.2

Multiply $x$ by $1$ .

$- 2 + x = 2$

Step 6.2

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

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Step 6.2.1

Add $2$ to both sides of the equation.

$x = 2 + 2$

Step 6.2.2

Add $2$ and $2$ .

$x = 4$