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Chapter 5: Problem 5

Write each expression with positive exponents only. Then simplify, if possible. $$(-6)^{-2}$$

Short Answer

Expert verified
So, the expression \( (-6)^{-2} \) with positive exponents and simplified is \( \frac{1}{36} \).

Step by step solution

01

Rewrite the expression with a positive exponent

By the rule of exponents stated in the analysis, \( (-6)^{-2} \) can be rewritten as \( \frac{1}{(-6)^2} \).
02

Simplify the expression

Now calculate \( (-6)^2 \) which equals 36. This makes the expression \( \frac{1}{36} \).

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Key Concepts

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

Exponents
Exponents are mathematical expressions used to describe repeated multiplication of a number by itself. In the expression \((-6)^{-2}\), the "-2" is the exponent, and it tells us how many times to multiply the base, which in this case is -6, by itself. But because it is a negative exponent, there is a twist to how it is handled.
When learning about exponents, it's essential to remember:
  • Any number raised to an exponent of 1 is the number itself: \( a^1 = a \).
  • Any number (except zero) raised to the exponent of 0 equals 1: \( a^0 = 1 \).
  • Exponents also follow associativity with multiplication: \( (a^m)^n = a^{m\cdot n} \).
Understanding exponents is a core part of algebra, allowing us to express large numbers compactly and solve mathematical problems efficiently.
Negative Exponents
Negative exponents can initially be confusing because they introduce the idea of division into exponentiation. The key thing to remember is that a negative exponent indicates the reciprocal of the base raised to a positive power. For instance, in the exercise \((-6)^{-2}\), the negative exponent changes to a reciprocal: \((-6)^{-2} = \frac{1}{(-6)^2}\).
Some important points about negative exponents include:
  • Negative exponents essentially "flip" the number: \( a^{-n} = \frac{1}{a^n} \).
  • They help in expressing fractions in a simpler form without using division.
  • When simplifying, always first introduce the reciprocal to change the exponent to positive.
Gaining comfort with negative exponents paves the way for simplifying expressions and understanding deeper algebraic concepts.
Simplifying Expressions
Simplifying expressions in algebra often involves rewriting them in a form that is easiest to work with. Simplifying allows you to clearly understand the essence of the expression and make future computations more manageable.
In our exercise, after rewriting the expression \((-6)^{-2}\) as \(\frac{1}{(-6)^2}\) using the rule for negative exponents, the next step is to simplify the expression by evaluating \((-6)^2\). Here, squaring -6 gives 36 because:
  • Squaring a negative results in a positive: \((-a)^2 = a^2 \).
  • After calculation, the expression becomes \(\frac{1}{36}\).
Simplification is crucial because it reduces complexity, allowing easier arithmetic operations and clearer interpretations of the expression. Always simplify as much as possible for best practice in algebra.

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Most popular questions from this chapter

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