Problem 54 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Simplify the expression, writing your answer using positive exponents only. $$ \left[\left(a^{-1}+b^{-1}\right)\left(a^{-1}-b^{-1}\right)\right]^{-2} $$

Step-by-Step Solution

Answer

The simplified expression, with only positive exponents, is \(a^4 - b^4\).

Step 1: Expand and simplify inside parentheses

We start by expanding the expression within brackets by multiplication: \[ \left(a^{-1}+b^{-1}\right)\left(a^{-1}-b^{-1}\right) = a^{-2} - b^{-2} \] Note that we multiplied the terms with their corresponding signs, this simplifies the expression inside the parentheses.

Step 2: Apply exponent rules to negative power

Now, we attach the outer exponent of -2 to the inner terms: \[ \left(a^{-2} - b^{-2}\right)^{-2} \] We have to raise each term inside the parentheses to the power of -2. \[ (a^{-2})^{-2} - (b^{-2})^{-2} \]

Step 3: Simplify the exponents

Using the rule that \(a^{m^n} = a^{m\cdot n}\), we can simplify the exponents: \[ a^{(-2)(-2)} - b^{(-2)(-2)} = a^4 - b^4 \] Now we have the expression in its simplest form, with only positive exponents.

Step 4: Write the final simplified expression

The final simplified expression, using only positive exponents, is: \[ a^4 - b^4 \]

Key Concepts

Negative Exponents

Negative exponents can initially seem confusing, but they're actually quite straightforward. A negative exponent indicates that the base should be inverted. For instance, if you have an expression like \( a^{-1} \), it is equivalent to \( \frac{1}{a} \). Think of the negative sign as a signal to "flip" the base to the bottom of a fraction. This idea helps make sense of expressions by changing negative exponents into positive ones.

Here’s a quick way to handle negative exponents:

• \( x^{-n} = \frac{1}{x^n} \)
• \( (\frac{1}{x})^{-n} = x^n \)

These rules are very handy when simplifying complex expressions, as shown in the exercise example where the expression \( a^{-1} + b^{-1} \) becomes easier to manage when thought of as \( \frac{1}{a} + \frac{1}{b} \). Understanding how to manipulate negative exponents allows for the reorganization of terms in a more workable form.

Exponent Rules

Dealing with exponents involves several fundamental rules that simplify calculations and expressions. These rules ensure consistency when multiplying, dividing, or raising powers within expressions.

Some essential exponent rules to remember:

• Product of Powers: \( x^m \cdot x^n = x^{m+n} \)
• Power of a Power: \( (x^m)^n = x^{m\cdot n} \)
• Quotient of Powers: \( \frac{x^m}{x^n} = x^{m-n} \) (for \( x eq 0 \))
• Power of a Product: \( (xy)^n = x^n \cdot y^n \)

In the given exercise, the rule \((x^m)^n = x^{m\cdot n}\) is crucial. This rule helps to simplify the expression \((a^{-2})^{-2} \) to \( a^{(-2)\cdot(-2)} \), resulting in \( a^4 \). By systematically applying these rules, one can substantially reduce the complexity of expressions and bring them to their simplest form.

Simplifying Expressions

Simplifying expressions often combines the application of negative exponents and exponent rules to achieve an expression with positive exponents, making them more manageable and easier to understand.

Steps for simplifying expressions:

• Identify and Apply Exponent Rules: Recognize which exponent rules apply and use them to condense the expression.
• Convert Negative Exponents: Turn any negative exponents into fractions, which makes further simplification easier.
• Simplify Fractions and Terms: Reduce fractions and combine like terms to streamline the expression.
• Final Check: Ensure that all exponents are positive, especially if specified in the problem.

In the exercise provided, we began with \( \left(a^{-1}+b^{-1}\right)\left(a^{-1}-b^{-1}\right) \). We expanded and simplified this to \( a^{-2} - b^{-2} \), then used the rule for multiplying exponents to convert those into \( a^4 - b^4 \). Mastering simplification requires practice but ultimately makes solving algebraic problems much more accessible.