Problem 10 - Prealgebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Substitute the value of a

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

Step 2: 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 \(\frac{1}{2^5}\).

Step 3: Calculate \(2^5\)

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

Step 4: 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 \(a=2\).

Key Concepts

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.