Problem 42 - Algebra for College Students

Question

Simplify. $$\frac{a^{-1}-b^{-1}}{a^{-1}+b^{-1}}$$

Step-by-Step Solution

Answer

\frac{b-a}{b+a}

Step 1 - Write the expression with positive exponents

Convert the negative exponents into positive exponents. Remember that \(a^{-1} = \frac{1}{a}\) and \(b^{-1} = \frac{1}{b}\). Hence, we can rewrite the given expression: \[ \frac{\frac{1}{a} - \frac{1}{b}}{\frac{1}{a} + \frac{1}{b}} \]

Step 2 - Find a common denominator for the fractions

Find a common denominator for the fractions in both the numerator and denominator of the expression. In this case, the common denominator is \(ab\). Rewrite each fraction: \[ \frac{\frac{b-a}{ab}}{\frac{b+a}{ab}} \]

Step 3 - Simplify the complex fraction

To simplify the complex fraction, divide the numerator by the denominator. This is done by multiplying the numerator by the reciprocal of the denominator: \[ \frac{b-a}{ab} \times \frac{ab}{b+a} \]

Step 4 - Cancel out common terms

Cancel out the common \(ab\) terms in the numerator and denominator: \[ \frac{b-a}{b+a} \]

Key Concepts

negative exponents

Negative exponents can be tricky, but they follow a simple rule. When you see a negative exponent, it means you take the reciprocal of the base and change the exponent to positive. For example, if you have \(a^{-1}\), you can rewrite it as \(\frac{1}{a}\). This is the first step in simplifying our expression \(\frac{a^{-1} - b^{-1}}{a^{-1} + b^{-1}}\). By changing the negative exponents to positive ones, we get:
\[ \frac{\frac{1}{a} - \frac{1}{b}}{\frac{1}{a} + \frac{1}{b}} \]
Switching negative exponents to positive exponents makes it easier to work with fractions, setting us up for the next steps.

complex fractions

Complex fractions have fractions within fractions. They look complicated, but don't worry! To simplify them, we need to find a common denominator for the smaller fractions first. When our expression \(\frac{\frac{1}{a} - \frac{1}{b}}{\frac{1}{a} + \frac{1}{b}}\) has fractions in both the numerator and the denominator, this makes it a complex fraction.
For these tasks, finding a common denominator helps to combine the fractions into a single fraction for both the numerator and the denominator. This approach keeps things less tangled and easier to handle.

common denominators

Finding a common denominator is essential for simplifying fractions. It allows you to combine them into a single fraction.
In our example, the common denominator for the mini-fractions \(\frac{1}{a}\) and \(\frac{1}{b}\) is \(ab\). You need this for both the numerator and the denominator in our complex fraction:
\[ \frac{\frac{b - a}{ab}}{\frac{b + a}{ab}} \]
With a common denominator, you can now combine the mini-fractions more quickly. This paves the way for the final steps in simplifying the expression.

fraction simplification

Simplifying fractions means making them as easy to understand as possible. After combining our mini-fractions with a common denominator, we are left with:
\[ \frac{b - a}{ab} \div \frac{b + a}{ab} \]
To simplify this, divide the numerator by the denominator by multiplying by the reciprocal of the denominator fraction: \[ \frac{b-a}{ab} \times \frac{ab}{b+a} \]
This cancels out the common \(ab\) terms: \[ \frac{b-a}{b+a} \]
And there you have it, the simplified form of the initial complex expression. Breaking down complex tasks into smaller steps makes the process more manageable.