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Chapter 4: Problem 10

Evaluate each expression if \(a=2\) and \(b=-3\) $$a^{-5}$$

Short Answer

Expert verified
\(a^{-5} = \frac{1}{32}\) when \(a=2\).

Step by step solution

01

Substitute the value of a

In the expression \(a^{-5}\), we substitute \(a = 2\). The expression becomes \(2^{-5}\).
02

Evaluate the expression

The expression \(2^{-5}\) means we take the reciprocal of \(2\) and raise it to the power of \(5\). Thus, it becomes \(\frac{1}{2^5}\).
03

Calculate \(2^5\)

We calculate \(2^5\) by multiplying \(2\) by itself five times: \(2 \times 2 \times 2 \times 2 \times 2 = 32\).
04

Solve the fraction

Now, we write the result as a fraction: \(\frac{1}{32}\). This is the final evaluation of the expression \(a^{-5}\) with \(a=2\).

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Key Concepts

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

Evaluating Expressions
Evaluating an expression is the process of finding its value when one or more variables are given specific values. Imagine expressions are like recipes with variables as ingredients. When you know what these ingredients are, you can "cook" and get a specific result. In math, we do this by replacing the variables with the given numbers and performing the necessary calculations.

For the expression given in the exercise, we have the expression with a negative exponent, specifically, \(a^{-5}\). With the value \(a = 2\), we plug it in to get \(2^{-5}\). This substitution is part of what makes up the entire evaluation process.
Substitution in Expressions
Substitution is like swapping puzzle pieces to complete a picture. In math, it's the act of replacing variables in an expression with known values. This step is crucial because it simplifies the expression and makes it possible to perform further calculations.

In our exercise, substitution required us to replace \(a\) with the given value of \(2\). So, \(a^{-5}\) transforms to \(2^{-5}\). Now, this expression can be understood as \(\frac{1}{2^5}\), which allows us to directly evaluate the power.
Powers of Numbers
When we talk about raising a number to a power, we mean multiplying it by itself a certain number of times, as indicated by the exponent. Powers make numbers express large multiplications easily and identify patterns and growth rates.

In this problem, we deal with \(2^5\), meaning \(2\) is multiplied by itself five times: \(2 \times 2 \times 2 \times 2 \times 2 = 32\). The negative exponent flips it into a fraction form: \(\frac{1}{32}\), emphasizing how negative exponents work as they convert regular powers into fractions, representing their reciprocal.

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Most popular questions from this chapter

Writing in Math Explain how simplified fractions are useful in representing measurements. Include an explanation of how measurements represent parts of a whole and examples of fractions that represent measurements.

Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. $$\frac{30}{37}$$

Find the greatest common factor of each set of numbers or monomials. $$42,54$$

Numbers can also be expressed in expanded form. Example \(1: 13,548=10,000+3000+500+40+8\) \(=\left(1 \times 10^{4}\right)+\left(3 \times 10^{3}\right)+\left(5 \times 10^{2}\right)+\left(4 \times 10^{1}\right)+\left(8 \times 10^{0}\right)\) Example \(2: 0.568=0.5+0.06+0.008\) $$ =\left(5 \times 10^{-1}\right)+\left(6 \times 10^{-2}\right)+\left(8 \times 10^{-3}\right) $$ Write each number in expanded form. $$5931$$

Find each equivalent measure. $$1 \text { day }=\underline{?} \mathrm{h}$$
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