Problem 43 - Beginning and Intermediate Algebra

Question

Evaluate. $$ (-2)^{4} $$

Step-by-Step Solution

Answer

The result of \((-2)^4\) is 16.

Step 1: Understanding the Base and Exponent

In the expression \((-2)^4\), the number \(-2\) is called the base and \(4\) is the exponent. The exponent tells us how many times the base is multiplied by itself.

Step 2: Multiplying the Base by Itself

To solve \((-2)^4\), multiply \(-2\) by itself 4 times: \((-2)\times(-2)\times(-2)\times(-2)\).

Step 3: First Multiplication

Calculate the first multiplication: \((-2) \times (-2) = 4\). Negative times negative gives a positive result.

Step 4: Second Multiplication

Use the result from the first multiplication in the next multiplication: \(4 \times (-2) = -8\). Positive times negative gives a negative result.

Step 5: Third Multiplication

Use the result from the second multiplication for the final multiplication: \(-8 \times (-2) = 16\). Negative times negative gives a positive result.

Step 6: Final Result

The final result of \((-2)^4\) is \(16\), which means \(-2\) multiplied by itself 4 times equals \(16\).

Key Concepts

Base

The base in a mathematical expression like \((-2)^4\) is the number that you repeatedly multiply. Here, the base is \(-2\). Understanding the base is essential because it tells you what number is being used throughout the process. In any expression of the form \(a^b\), \(a\) is the base, and it is the number you start with and keep multiplying. Using a base, especially a negative one like \(-2\), involves special rules in multiplication that we will explore further.

• The base determines the size and sign of the number.
• In expressions, the base is the starting point.

Whenever you identify a base, pay attention to its sign and size, as both play a role in the final outcome when you multiply it multiple times.

Exponentiation

Exponentiation is the process of taking a base and multiplying it by itself a certain number of times. It is expressed in the format \(a^b\), where \(b\) is the exponent. In our example, \((-2)^4\), the exponent is \(4\). This tells you to multiply the base \(-2\) by itself four times.
Understanding exponentiation involves the following key points:

• The exponent tells you the number of times to use the base in multiplication.
• A higher exponent means more repeated multiplication, increasing the power and size of the final number.
• Exponentiation is a form of repeated multiplication.

In essence, exponentiation is a shortcut to express repeated multiplication efficiently. Instead of writing \((-2) \times (-2) \times (-2) \times (-2)\), you simply write \((-2)^4\).

Negative Numbers

Negative numbers introduce unique properties when involved in exponentiation. In our problem, the base \(-2\) is a negative number. When multiplied, negative numbers produce interesting outcomes:

• Negative times negative results in a positive number.
• Negative times positive results in a negative number.
• Understanding the number of multiplications is crucial as it affects the sign of the result.

In the operation \((-2)^4\), you multiply four times:- The first multiplication yields positive \(4\).- The second multiplication resulting in negative \(-8\).- The third multiplication returns a positive \(16\) again.This consistent toggle between signs is an important characteristic of working with negative bases in exponentiation.

Multiplication Process

When dealing with exponentiation and negative bases, understanding the multiplication process helps in calculating accurately. We follow a step-by-step approach:
Each multiplication step follows a consistent rule, especially when negative numbers are involved:

• The first step in the multiplication process involves multiplying the base by itself. For \((-2)^4\), that is \((-2) \times (-2) = 4\)
• The second multiplication then uses the result from the first multiplication: \(4 \times (-2) = -8\)
• The third multiplication continues likewise to end up with a positive \(16\)

Every step in this process is essential to reaching the correct result. Understanding the switch between positive and negative results also enhances your grasping of this multiplication process, especially when dealing with negative bases.