Problem 45 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Simplify the expression, writing your answer using positive exponents only. $$ \left[\left(\frac{a^{-2} b^{-2}}{3 a^{-1} b^{2}}\right)^{2}\right]^{-1} $$

Step-by-Step Solution

Answer

The simplified expression using positive exponents only is \(\frac{ab^4}{9}\).

Step 1: Apply the power of a quotient rule

Recall that for any nonzero number \(a\), \(\frac{a^m}{a^n} = a^{m-n}\). We will apply this rule to our given expression: $$ \left[\left(\frac{a^{-2} b^{-2}}{3 a^{-1} b^{2}}\right)^{2}\right]^{-1} $$ Step 2: Simplify the rational expression using exponent rules

Step 2: Simplify using exponent rules

Now, compute the new exponents of \(a\) and \(b\) separately: $$ a^{-2-(-1)}b^{-2-2}\cdot 3^2 = a^{-1}b^{-4}\cdot 9 $$ Step 3: Square the expression

Step 3: Square the expression

Apply \((a^m)^n = a^{m \cdot n}\) to the simplified expression: $$ \left(a^{-1}b^{-4}\cdot 9\right)^{-1} $$ Step 4: Invert and simplify the expression

Step 4: Invert and simplify

Apply the exponent rules to get rid of the negative exponent by inverting the expression and change the sign of the exponent: $$ \frac{1}{a^{-1}b^{-4}\cdot 9} = \frac{1}{a^{-1}b^{-4}\cdot 9} \cdot \frac{a^{1}b^{4}}{a^{1}b^{4}} = \frac{a^{1}b^{4}}{9} $$ Step 5: Final answer

Step 5: Write the final answer

We have simplified the expression and written it using positive exponents only: $$ \frac{ab^4}{9} $$

Key Concepts

Simplifying Expressions

Simplifying expressions involves breaking down complex mathematical terms into simpler, more manageable forms. This often includes using various mathematical rules and operations to combine or reduce terms.

In the exercise provided, our goal is to simplify the expression entirely, preferably rewriting it with positive exponents. We often encounter expressions with exponents, fractions, or variables raised to various powers. Using the rules of exponents, we can systematically simplify these expressions.

Here, we begin by rewriting the given expression, apply rules to simplify each part step by step, and ensure at each stage that no ambiguous terms remain. While simplifying, it's common to stay vigilant about maintaining the balance of the equation, ensuring both sides are equivalent as we manipulate terms.

Power of a Quotient Rule

The Power of a Quotient Rule is a helpful tool when simplifying complex expressions involving division. According to this rule, for any nonzero numbers, if you have a fraction raised to an exponent, such as \( \left(\frac{a}{b}\right)^n \), it equals \( \frac{a^n}{b^n} \).

In the provided exercise, this principle helps to quickly manage the powers on both the numerator and the denominator. First, you simplify each portion of the fraction:

• Simplify \( \frac{a^{-2} b^{-2}}{3 a^{-1} b^{2}} \) using the power of a quotient rule, by subtracting the exponents of alike bases both in the numerator and denominator.
• After simplification, apply the same power to both parts of the fraction as per the rules of exponents.

Consequently, this rule makes dealing with fractions less daunting and results in a streamlined expression allowing further simplification.

Negative Exponents

Negative exponents can initially be confusing, but understanding them can dramatically simplify expressions. When you see a negative exponent, think of it as a direction on whether a term is part of the numerator or denominator.

For instance, a negative exponent like \( a^{-n} \) is equivalent to \( \frac{1}{a^n} \). This means that terms with negative exponents can be flipped in the fraction to eliminate the negativity.

In solving our exercise, we use this principle to eliminate negative exponents:

• The negative exponents in the expression \( a^{-1}b^{-4} \) after inversion become positive when transferred to the numerator, forming \( a^1b^4 \).
• Couple this with the constant in the expression \( 9 \), and you finally achieve an expression with all positive exponents.

By understanding and applying this key concept, we take away the complexity from expressions involving negative exponents, making it efficient to solve.