Precalculus Examples

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

Step 1

Apply the product rule to $\frac{1}{25}$ .

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

Step 2

One to any power is one.

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

Step 3

Move $25^{\frac{1}{x}}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 4

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

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

Step 5

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

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

Step 6

Solve for $x$ .

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Step 6.1

Simplify $2 (- \frac{1}{x})$ .

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Step 6.1.1

Multiply $2 (- \frac{1}{x})$ .

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Step 6.1.1.1

Multiply $- 1$ by $2$ .

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

Step 6.1.1.2

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

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

Step 6.1.2

Move the negative in front of the fraction.

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

Step 6.2

Find the LCD of the terms in the equation.

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Step 6.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x, 1$

Step 6.2.2

The LCM of one and any expression is the expression.

$x$

Step 6.3

Multiply each term in $- \frac{2}{x} = 1$ by $x$ to eliminate the fractions.

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Step 6.3.1

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

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

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $x$ .

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Step 6.3.2.1.1

Move the leading negative in $- \frac{2}{x}$ into the numerator.

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

Step 6.3.2.1.2

Cancel the common factor.

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

Step 6.3.2.1.3

Rewrite the expression.

$- 2 = 1 x$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Multiply $x$ by $1$ .

$- 2 = x$

Step 6.4

Rewrite the equation as $x = - 2$ .

$x = - 2$