Problem 60 - Algebra for College Students

Question

Simplify. $$\left(a^{-1}-b^{-1}\right)^{-2}$$

Step-by-Step Solution

Answer

\(\left( \frac{ab}{b - a} \right)^2\)

Step 1: Rewrite negative exponents

First, rewrite the negative exponents inside the parentheses using the property \(a^{-n} = \frac{1}{a^n}\).This gives \((\frac{1}{a} - \frac{1}{b})^{-2}\).

Step 2: Find a common denominator

Combine the fractions inside the parentheses by finding a common denominator, which is \(ab\). The expression becomes \((\frac{b - a}{ab})^{-2}\).

Step 3: Apply the negative exponent rule

Apply the negative exponent rule \((\frac{m}{n})^{-p} = (\frac{n}{m})^p\) to the expression \((\frac{b - a}{ab})^{-2}\). This gives \((\frac{ab}{b - a})^2\).

Step 4: Simplify

The simplified form of the expression is \(\left( \frac{ab}{b - a} \right)^2\).

Key Concepts

Negative Exponents

Negative exponents can seem tricky at first, but they are quite straightforward once you get the hang of them. A negative exponent indicates that the base (the number being raised to the exponent) should be taken as a reciprocal. So, the expression: \(a^{-1}\) can be rewritten as \(\frac{1}{a}\).

This means that for any number or variable \(x\), \(x^{-n} = \frac{1}{x^n}\). In our exercise, we applied this rule and changed \(a^{-1}\) and \(b^{-1}\) to \(\frac{1}{a}\) and \(\frac{1}{b}\) respectively. This helps make our expressions simpler and easier to work with.

Remember, handling negative exponents is all about flipping the base to the denominator.

Common Denominator

When dealing with fractions inside expressions, it's often necessary to combine them into a single fraction. To do that, we need a common denominator. A common denominator is a number that can evenly divide all the denominators in your set of fractions.

For our exercise, we had \(\frac{1}{a}\) and \(\frac{1}{b}\). The denominators are \(a\) and \(b\). The least common multiple of \(a\) and \(b\) is simply \(ab\).

Now, we rewrite the fractions with this common denominator: \[ \frac{b}{ab} - \frac{a}{ab} = \frac{b - a}{ab} \].

Combining the fractions makes them easier to manipulate in future steps.

Exponent Rules

Exponent rules are vital for simplifying algebraic expressions. One important rule is the negative exponent rule, which we used in the first step. The negative exponent rule states that \( (\frac{m}{n})^{-p} = (\frac{n}{m})^p \). Applying this rule helps inverting and simplifying complex fractions.

Another important concept is that when you raise a fraction to a negative exponent, you flip the fraction and make the exponent positive. Like in our exercise: \[ (\frac{(b - a)}{ab})^{-2} = (\frac{ab}{(b - a)})^2 \].

Mastering these rules makes simplifying expressions more manageable and helps to approach problems effectively.

Fraction Simplification

Fractions often appear in algebraic expressions, and simplifying them is crucial. Simplification involves reducing a fraction to its simplest form and handling any exponents.

In our exercise, we ended with \((\frac{ab}{b - a})^2\). This is a clear and simplified version of the original complex expression. We used several steps, involving negative exponents and common denominators, to achieve this form.

Understanding the process of combining, reducing, and manipulating fractions will enhance your ability to solve algebraic equations efficiently.