Pre-Algebra Examples

$x = {(\frac{a}{b})}^{\frac{1}{n}} - 1$

Step 1

Rewrite the equation as ${(\frac{a}{b})}^{\frac{1}{n}} - 1 = x$ .

${(\frac{a}{b})}^{\frac{1}{n}} - 1 = x$

Step 2

Apply the product rule to $\frac{a}{b}$ .

$\frac{a^{\frac{1}{n}}}{b^{\frac{1}{n}}} - 1 = x$

Step 3

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

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Step 3.1

Add $1$ to both sides of the equation.

$\frac{a^{\frac{1}{n}}}{b^{\frac{1}{n}}} - x = 1$

Step 3.2

Add $x$ to both sides of the equation.

$\frac{a^{\frac{1}{n}}}{b^{\frac{1}{n}}} = 1 + x$

Step 4

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (\frac{a^{\frac{1}{n}}}{b^{\frac{1}{n}}}) = \ln (1 + x)$

Step 5

Expand the left side.

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Step 5.1

Rewrite $\ln (\frac{a^{\frac{1}{n}}}{b^{\frac{1}{n}}})$ as $\ln (a^{\frac{1}{n}}) - \ln (b^{\frac{1}{n}})$ .

$\ln (a^{\frac{1}{n}}) - \ln (b^{\frac{1}{n}}) = \ln (1 + x)$

Step 5.2

Expand $\ln (a^{\frac{1}{n}})$ by moving $\frac{1}{n}$ outside the logarithm.

$\frac{1}{n} \ln (a) - \ln (b^{\frac{1}{n}}) = \ln (1 + x)$

Step 5.3

Expand $\ln (b^{\frac{1}{n}})$ by moving $\frac{1}{n}$ outside the logarithm.

$\frac{1}{n} \ln (a) - (\frac{1}{n} \ln (b)) = \ln (1 + x)$

Step 5.4

Combine $\frac{1}{n}$ and $\ln (a)$ .

$\frac{\ln (a)}{n} - (\frac{1}{n} \ln (b)) = \ln (1 + x)$

Step 5.5

Combine $\frac{1}{n}$ and $\ln (b)$ .

$\frac{\ln (a)}{n} - \frac{\ln (b)}{n} = \ln (1 + x)$

Step 6

Simplify the left side.

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Step 6.1

Simplify $\frac{\ln (a)}{n} - \frac{\ln (b)}{n}$ .

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Step 6.1.1

Combine the numerators over the common denominator.

$\frac{\ln (a) - \ln (b)}{n} = \ln (1 + x)$

Step 6.1.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\frac{\ln (\frac{a}{b})}{n} = \ln (1 + x)$

Step 7

Find the LCD of the terms in the equation.

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Step 7.1

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

$n, 1$

Step 7.2

The LCM of one and any expression is the expression.

$n$

Step 8

Multiply each term in $\frac{\ln (\frac{a}{b})}{n} = \ln (1 + x)$ by $n$ to eliminate the fractions.

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Step 8.1

Multiply each term in $\frac{\ln (\frac{a}{b})}{n} = \ln (1 + x)$ by $n$ .

$\frac{\ln (\frac{a}{b})}{n} n = \ln (1 + x) n$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $n$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{\ln (\frac{a}{b})}{n} n = \ln (1 + x) n$

Step 8.2.1.2

Rewrite the expression.

$\ln (\frac{a}{b}) = \ln (1 + x) n$

Step 9

Solve the equation.

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Step 9.1

Rewrite the equation as $\ln (1 + x) n = \ln (\frac{a}{b})$ .

$\ln (1 + x) n = \ln (\frac{a}{b})$

Step 9.2

Divide each term in $\ln (1 + x) n = \ln (\frac{a}{b})$ by $\ln (1 + x)$ and simplify.

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Step 9.2.1

Divide each term in $\ln (1 + x) n = \ln (\frac{a}{b})$ by $\ln (1 + x)$ .

$\frac{\ln (1 + x) n}{\ln (1 + x)} = \frac{\ln (\frac{a}{b})}{\ln (1 + x)}$

Step 9.2.2

Simplify the left side.

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Step 9.2.2.1

Cancel the common factor of $\ln (1 + x)$ .

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Step 9.2.2.1.1

Cancel the common factor.

$\frac{\ln (1 + x) n}{\ln (1 + x)} = \frac{\ln (\frac{a}{b})}{\ln (1 + x)}$

Step 9.2.2.1.2

Divide $n$ by $1$ .

$n = \frac{\ln (\frac{a}{b})}{\ln (1 + x)}$