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Chapter 33: Problem 948

Simplify the following expressions: (a) \(-3^{-2}\) (b) \((-3)^{-2}\) (c) \((-3) /\left(4^{-1}\right)\)

Short Answer

Expert verified
(a) \(\frac{1}{9}\) (b) \(\frac{1}{9}\) (c) \(-12\)

Step by step solution

01

(a) Simplify \(-3^{-2}\)

First, we need to understand the meaning of a negative exponent. A negative exponent signifies that we should take the reciprocal of the base, which is the number with the exponent. In this case, the base is -3. So, we can rewrite the expression as follows: \[ \frac{1}{(-3)^{2}} \] Now, we can compute the power of 2 for -3. \[ \frac{1}{9} \] So, \(-3^{-2}\) simplifies to \(\frac{1}{9}\).
02

(b) Simplify \((-3)^{-2}\)

In this case, the negative sign is within the parentheses which means the entire expression (-3) should be taken to the power of -2. \[ \frac{1}{(-3)^{2}} \] Compute the power of 2 for -3. \[ \frac{1}{9} \] Therefore, \((-3)^{-2}\) simplifies to \(\frac{1}{9}\), just like in the previous part.
03

(c) Simplify \(\frac{-3}{4^{-1}}\)

First, we need to take the reciprocal of 4 due to the negative exponent (-1). \[ \frac{-3}{\left(\frac{1}{4}\right)} \] Now we can simplify the expression by multiplying the numerator and the denominator by 4: \[ -3 \times 4 = -12 \] \[ \frac{1}{4} \times 4 = 1 \] Thus, the simplified expression is: \(\frac{-12}{1}\), which is equal to \(-12\).

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Key Concepts

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

Negative Exponents
Negative exponents might seem confusing at first, but they follow a simple rule: turn the number with the exponent upside down! That is, a negative exponent tells you to take the reciprocal (or the 'flipped version') of the number.
For instance, if you have a base number with a negative exponent, like \(-3^{-2}\), the negative exponent indicates inversion. You can rewrite the expression as \(\frac{1}{(-3)^2}\). This rule applies to any negative exponent, regardless of the base number.
  • A negative exponent inverts the base.
  • Turn the base into a fraction: \(a^{-n} = \frac{1}{a^n}\).
Reciprocals
Reciprocals are quite important in mathematics, especially when dealing with exponents. The reciprocal of a number is simply 1 divided by that number.
For example, the reciprocal of 4 is \(\frac{1}{4}\), and the reciprocal of \(\frac{1}{4}\) is 4. This is because 4 times \(\frac{1}{4}\) equals 1.
  • Flipping numerators and denominators gives you reciprocals.
  • If \(a = \frac{1}{b}\), then \(b = \frac{1}{a}\).

In expressions like \(\frac{-3}{4^{-1}}\), simply change \(4^{-1}\) to its reciprocal form: \(\frac{1}{4}\). Afterwards, multiplying by 4 simplifies the fraction.
Simplifying Expressions
Simplifying expressions means reducing them to their simplest, cleanest form. This could mean combining like terms, factoring, or using other mathematical properties.
  • The goal: fewer operations and simpler numbers.
  • Follow mathematical rules of order to make expressions manageable.

In this exercise, simplifying \(3^{-2}\) involved turning it into \(\frac{1}{9}\) by resolving negative exponents and computing the power. Our expression for the negative exponent situations became much clearer after simplification.
When dealing with division, like \(\frac{-3}{4^{-1}}\), simplifying eventually led to \(-12\). You encountered fractions and conversion by reciprocals.
Power of Numbers
Taking a number to a power means multiplying the number by itself a certain number of times. For example, \(3^2\) means \(3 \times 3\), which is 9.
  • \(a^n\) means multiplying \(a\) by itself \(n\) times.
  • Common powers: squares \((^2)\), cubes \((^3)\), etc.

In the exercises, you calculated the power of -3 by squaring it to get 9 in both examples. Understanding powers is crucial because many math problems demand these calculations.
Developing a strong grasp on this topic expands your math toolset, opening doors to solve more complex problems involving exponents.

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