Problem 34 - Precalculus: Functions and Graphs

Question

Simplify the expression. $$\left(a^{-1}+b^{-1}\right)^{-1}$$

Step-by-Step Solution

Answer

\( \frac{ab}{a+b} \)

Step 1: Rewrite the Expression

The original expression is \( \left(a^{-1} + b^{-1}\right)^{-1} \). To simplify, we first rewrite \( a^{-1} \) as \( \frac{1}{a} \) and \( b^{-1} \) as \( \frac{1}{b} \). Therefore, the expression becomes \( \left( \frac{1}{a} + \frac{1}{b} \right)^{-1} \).

Step 2: Find a Common Denominator

To simplify \( \frac{1}{a} + \frac{1}{b} \), we need a common denominator. The common denominator for \( a \) and \( b \) is \( ab \). Using this, rewrite the terms: \( \frac{1}{a} = \frac{b}{ab} \) and \( \frac{1}{b} = \frac{a}{ab} \).

Step 3: Combine the Fractions

Combine the fractions: \( \frac{1}{a} + \frac{1}{b} = \frac{b}{ab} + \frac{a}{ab} = \frac{a + b}{ab} \).

Step 4: Inverse the Fraction

Now we take the inverse of the fraction \( \left( \frac{a+b}{ab} \right)^{-1} \). The inverse of a fraction \( \frac{x}{y} \) is \( \frac{y}{x} \). Thus, \( \left( \frac{a+b}{ab} \right)^{-1} = \frac{ab}{a+b} \).

Key Concepts

Fraction Addition

When you encounter a situation where two fractions need to be added together, it's essential to remember that the denominators must be the same. If the denominators are not the same, like in the fractions \( \frac{1}{a} \) and \( \frac{1}{b} \), you'll need to find a common denominator before adding them together. This process ensures that you can successfully combine the fractions into a single expression.
To add the fractions \( \frac{1}{a} \) and \( \frac{1}{b} \), you need to:

• Find the least common denominator (LCD) by calculating the smallest multiple that is common to both denominators \(a\) and \(b\), which is simply \(ab\).
• Rewrite each fraction with this common denominator. So, \(\frac{1}{a} = \frac{b}{ab}\) and \(\frac{1}{b} = \frac{a}{ab}\).
• Add the numerators: \( \frac{b}{ab} + \frac{a}{ab} = \frac{a+b}{ab} \).

By following these steps, you'll effectively add the fractions. This basic understanding of fraction addition is crucial in algebra, particularly when further simplification of expressions is necessary.

Inverse of a Fraction

The concept of the inverse, often referred to as the reciprocal, of a fraction is key in algebra. It involves flipping the fraction, which means swapping its numerator and denominator. This operation is particularly useful when you encounter negative exponents, as they can be converted into positive exponents by using this notion.
For example, if you have the expression \( \left( \frac{a+b}{ab} \right)^{-1} \), finding the inverse would mean you flip the fraction to get \( \frac{ab}{a+b} \).

• The inverse plays a crucial role in dividing fractions and solving equations where you need to isolate variables.
• It’s beneficial to visualize this operation as simply flipping the fraction to help in understanding its effect on the equation.

Understanding the inverse operation expands your ability to manipulate and simplify complex algebraic expressions by making use of the fraction’s reciprocal.

Common Denominator

In any algebraic manipulation involving fractions, identifying a common denominator is a vital step. It allows you to rewrite fractions in such a way that they have the same base, making addition, subtraction, and comparison simple and straightforward.
For instances involving different denominators like \(\frac{1}{a}\) and \(\frac{1}{b}\), here's why you find a common denominator:

• The common denominator \( ab \) can be visualized as a 'bridge' that allows these fractions to be combined.
• By ensuring the fractions have a common denominator, \( \frac{1}{a} \) becomes \( \frac{b}{ab} \), and \( \frac{1}{b} \) becomes \( \frac{a}{ab} \).
• This helps further operations like addition to occur seamlessly, combining to \( \frac{a+b}{ab} \).

A common denominator simplifies algebraic expressions and is one of the fundamental strategies used in solving problems involving fractions. It also lays the groundwork for working with algebraic equations efficiently.

Negative Exponents

Negative exponents are an expression of how exponents reflect repeated division, just as regular exponents indicate repeated multiplication. A negative exponent means that instead of multiplying by the base, you divide by it. Consequently, it transforms the exponentiation into its reciprocal form.
Take, for instance, the transformation of \( a^{-1} \) into \( \frac{1}{a} \). This switch from a negative exponent to a fractional notation reflects that negative exponents essentially point towards the inverse.

• Negative exponents translate into reciprocal terms, hence \( x^{-n} = \frac{1}{x^n} \).
• It is a pivotal understanding in algebra that helps tackle expressions like \( \left(a^{-1} + b^{-1} \right)^{-1} \) by converting terms into fractions.
• By knowing the rules of negative exponents, you can handle different algebraic scenarios efficiently, making simplifications and solving equations less daunting.

Mastering negative exponents unlocks the ability to simplify and rearrange complex algebraic expressions, allowing for more straightforward solutions.