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Chapter 9: Problem 83

For Exercises, simplify. $$\left(-2 a b^{-2}\right)\left(4 a^{-2} b\right)^{-2}$$

Short Answer

Expert verified
The simplified form of the expression \(\left(-2 a b^{-2}\right)\left(4 a^{-2} b\right)^{-2}\) is \(\frac{-a^{4}}{128b^{4}}\).

Step by step solution

01

Understand the Expression

First, notice that the expression is a single term expression involving variables \(a\) and \(b\) with negative exponents. Before you proceed, understand that a negative exponent can be changed into a positive exponent by moving the term from numerator to denominator or from denominator to numerator. Thus, the expression can be rewritten as \(-2a(b^{2})^{-1}\) * \((4a^{-2}b)^{-2}\).
02

Simplify Negative Exponents

Apply the rule to change the negative exponent to positive by moving \((b^{2})^{-1}\) from numerator to denominator and \((4a^{-2}b)^{-2}\) from numerator to denominator (or vice-versa) giving \(-2a/(b^{2}) * 1/(4^{2}a^{-4}b^{2})\). This simplifies to \(-2a/(b^{2}) * 1/(16a^{-4}b^{2})\).
03

Simplify the Expression

Now multiply both terms togther and simplify as per exponent rule by subtracting the exponent of \(a\) and \(b\) in numerator and denominator respectively to get the final answer. Which gives \(-2a/(256a^{-3}b^{4}) = -2a^{4}/(256b^{4}) = -a^{4}/(128b^{4})\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Negative Exponents
Negative exponents might seem a bit tricky at first, but they are quite simple once you understand the concept. The main idea is that a negative exponent indicates that the number should be moved from one part of the fraction to the other. This means:
  • If you have a term with a negative exponent in the numerator, you move it to the denominator and make the exponent positive.
  • If the term is in the denominator with a negative exponent, bring it up to the numerator with a positive exponent.
For example, converting a negative exponent to positive: \[ x^{-n} = \frac{1}{x^n} \]By doing this, the base remains the same, but the exponent loses its negative sign. Knowing how to handle negative exponents will help you simplify expressions fully.
The Fundamental Exponent Rules
There are several rules in algebra that make working with exponents straightforward. These rules are essential when simplifying expressions involving exponents:
  • Product of Powers Rule: When you multiply terms with the same base, add their exponents: \( a^m \times a^n = a^{m+n} \)
  • Power of a Power Rule: When raising an exponent to another exponent, multiply the exponents: \( (a^m)^n = a^{m \times n} \)
  • Quotient of Powers Rule: When dividing terms with the same base, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \)
These rules apply regardless of whether the exponents are positive or negative. Properly applying these rules allows you to manipulate and simplify expressions efficiently. Keep these fundamental rules in mind as you work through algebraic problems.
Working with Algebraic Expressions
Algebraic expressions, like the given example, involve terms that can include numbers, variables, and exponents. Understanding how to simplify such expressions is crucial for solving more complex algebra problems.
To simplify algebraic expressions, follow these steps:
  • Look for terms with similar bases so you can use exponent rules for simplification.
  • Convert negative exponents to positive by moving parts of the expression between numerator and denominator.
  • Once exponents are positive, apply the exponent rules discussed earlier to combine and simplify terms.
The goal is to reduce the expression to its simplest form, ideally with no negative exponents and the fewest terms possible. Practice applying these techniques to become efficient at tackling any algebraic problem you might face.

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