Problem 2 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Evaluate the expression. $$ (-2)^{5} $$

Step-by-Step Solution

Answer

The value of the expression \((-2)^5\) is -32.

Step 1: Identify the base and exponent

In the given expression, \((-2)^5\), the base is -2 and the exponent is 5.

Step 2: Apply the rules of exponents

Since the exponent is an odd number, raising a negative number to an odd power will result in a negative value. In this case: \[ (-2)^5 = -2 \times -2 \times -2 \times -2 \times -2 \]

Step 3: Calculate the value

Now, we will multiply -2 five times to find the value of the expression: \[ -2 \times -2 = 4 \] \[ 4 \times -2 = -8 \] \[ -8 \times -2 = 16 \] \[ 16 \times -2 = -32 \] So, the value of the expression \((-2)^5 = -32\).

Key Concepts

Negative Bases and Exponents

Understanding how negative bases and exponents work is essential in grasping fundamental mathematical concepts. When working with negative bases and positive exponents, the result depends on whether the exponent is even or odd. With an even exponent, the negative base will be multiplied by itself an even number of times, always resulting in a positive value. For instance, \( (-2)^2 = 4 \). However, when the exponent is odd, as in \( (-2)^3 \), we are multiplying the negative base an odd number of times, which results in a negative value, so \( (-2)^3 = -8 \).

It's crucial to remember that the rules change when dealing with negative exponents. A negative exponent signifies division, or more specifically, taking the reciprocal of the base raised to the corresponding positive exponent. For example, \( (-2)^{-1} = -\frac{1}{2} \) because you are effectively dividing 1 by \( -2 \). This might seem tricky at first, but with practice, determining the results of negative bases combined with different exponents becomes more intuitive.

Odd Power Exponentiation

When we talk about odd power exponentiation, we're looking at what happens when we raise any number to an exponent that is an odd number. As seen in the example \( (-2)^5 \), we multiply the base, \( -2 \) by itself five times because the exponent is 5—clearly an odd number. The critical takeaway for odd powers is that they retain the sign of the base. So, raising a negative number to an odd exponent results in a negative number, as in this case with \( -32 \). Similarly, a positive number raised to an odd power will always yield a positive result.

This particular concept is important when solving various mathematical problems, such as polynomial equations, where the power of variables can significantly affect the sign and thus the direction of a graph. Recognizing the simple pattern that odd exponents maintain the sign of the base can be a quick aid in mental calculations and algebraic manipulations.

Evaluating Expressions

Evaluating expressions is a fundamental skill in algebra. To evaluate an expression, follow the proper order of operations and apply the relevant rules for exponents, multiplication, division, addition, and subtraction. When dealing with expressions that include exponents, calculate them first before performing other operations. For example, in the expression \( (-2)^5 \), there are no other operations to perform besides exponentiation, so we focus solely on multiplying \( -2 \), five times, as demonstrated in the solution steps.

For more complex expressions involving both exponents and other operations, you might have to distribute, combine like terms, or factor as well. Evaluating expressions carefully by staying organized and double-checking each step can prevent errors, leading to a better understanding of algebraic processes and the correct solution.