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Chapter 1: Problem 60

In Exercises \(57-68,\) perform the indicated operations. Is \(a^{-2}

Short Answer

Expert verified
Yes, for any negative value of \(a\), \(a^{-2}\) is less than \(a^{-1}\).

Step by step solution

01

Understanding the Problem

We need to analyze the inequality \(a^{-2} < a^{-1}\) for negative values of \(a\). Since \(a\) is negative, we should consider what happens to negative bases when raised to negative powers.
02

Simplifying Negative Powers

Recall that \(a^{-n} = \frac{1}{a^n}\). For \(a^{-2}\), it becomes \(\frac{1}{a^2}\), and for \(a^{-1}\), it becomes \(\frac{1}{a}\). Thus the inequality \(a^{-2} < a^{-1}\) translates to \(\frac{1}{a^2} < \frac{1}{a}\).
03

Comparing Fractional Values

Since \(a\) is negative, \(a^2 > a\). When the base \(a\) is squared, the absolute value becomes larger, making \(a^2\) positive. As \(|a^2| > |a|\), it follows that \(\frac{1}{a^2}\) is less than \(\frac{1}{a}\) because the reciprocal of a larger positive number is smaller. Thus, \(\frac{1}{a^2} < \frac{1}{a}\) holds true.
04

Conclusion

For any negative value of \(a\), \(a^{-2}\) is indeed less than \(a^{-1}\), meaning \(a^{-2} < a^{-1}\) is always true when \(a\) is negative.

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Key Concepts

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

Understanding Inequalities with Negative Exponents
In mathematics, inequalities help us compare the size or order of two values, teaching us how different expressions relate to each other. When dealing with inequalities involving negative exponents, it involves understanding how these exponents affect the values.

Negative exponents, like in the inequality \(a^{-2}
To grasp it better:
  • Subtracting exponents while dividing powers might seem counterintuitive, yet it's crucial in comparing such expressions.
  • Understanding the implications of these operations for negative values is vital since squaring negative numbers results in positive outcomes.
Fraction Comparison with Negative Values
Fraction comparison with negative numbers can be tricky due to the nature of negative and positive values. When dealing with inequalities such as \(\frac{1}{a^2} < \frac{1}{a}\), it's important to analyze how negative values of \(a\) affect the equation.

Let's break down the process:
  • Negative bases squared turn positive. Thus, for any negative \(a\), \(a^2\) becomes a smaller (more negative) value than \(a\).
  • Although \(a^2
When comparing fractions for any negative \(a\), we find \(\frac{1}{a^2}<\frac{1}{a}\). Therefore, in this case, the inequality \(a^{-2}
Exploring Negative Bases with Exponents
Negative bases introduce unique challenges, particularly when exponents are involved. It's essential to comprehend their behavior in powers to ace problems like \(a^{-2}
Key points when exploring negative bases include:
  • Negative numbers serve as particular instances where their absolute values exceed their original values after squaring (e.g., for \(a = -3\), \(a^2 = 9\), a positive result).
  • When raised to negative exponents, negative bases affect inequality directions due to deriving reciprocal values.
This understanding ensures that when solving inequalities and comparing fractions involving negative numbers, the outcomes conform to the expected properties. Ultimately, by recognizing these patterns, solving expressions with negative bases and different exponents becomes a breeze.

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