Problem 108 - Elementary Algebra

Question

Your friend keeps getting an answer of 16 when simplifying \(-2^{4}\). What mistake is he making and how would you help him?

Step-by-Step Solution

Answer

The correct answer is -16; the negative sign is not part of the exponentiation.

Step 1: Understanding the Expression

The expression \(-2^4\) needs to be broken down correctly. It's important to understand that the minus sign is not being raised to the power of four.

Step 2: Application of Exponent Rules

According to exponent rules, \(-2^4\) means \(-(2^4)\). This indicates that only the 2 is affected by the power, and the negative sign is separate.

Step 3: Amplifying the Correct Calculation

Raise the base 2 to the power of 4: \(2^4 = 16\). Only 2 is raised to the power of 4.

Step 4: Applying the Negative Sign

Now, apply the negative sign to the result obtained from the power calculation: \(- (16) = -16\). Therefore, \(-2^4 = -16\).

Key Concepts

Negative Numbers

Negative numbers are numbers less than zero, commonly indicated by a minus sign in front of them. When working with negative numbers, it's crucial to understand how they behave in mathematical operations. For instance, multiplying, dividing, or considering powers of negative numbers can yield different results.

In our example, the expression \(-2^4\) is evaluated as \(-(2^4)\), meaning only the number 2 is raised to the fourth power, and the negative sign is simply a prefix. This negation results in a negative answer, highlighting how important it is to distinguish between operating on negative numbers and numbers raised to an exponent.

• Negative numbers change the sign of the outcome when multiplied by an odd number of factors.
• When multiplied by an even number of negative factors, the result is positive.

Understanding negative numbers is key to applying algebraic rules correctly and avoiding common mistakes.

Powers of Numbers

The concept of powers involves multiplying a number by itself a specified number of times. The power, or exponent, tells us how many times to use the base in a multiplication.For example, \(2^4\) means \(2 \times 2 \times 2 \times 2 = 16\). Here, 2 is the base, and 4 is the exponent.

Key points about powers:

• The base number is the number being multiplied repeatedly.
• The exponent indicates the number of times the base is multiplied by itself.
• Powers of numbers become significant in computations involving large numbers and algebraic expressions.

Being clear about what parts of the expression are included in the exponentiation leads to more accurate results and a better understanding of the problem.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They allow us to generalize mathematical concepts and solve problems using symbolic representations. Understanding how to manipulate these expressions is crucial in both basic and advanced algebra.

When working with expressions like \(-2^4\), it's important to interpret the notation as intended. Algebraic expressions can often have implied meanings or conventions that dictate order of operations.

• Parentheses can change the precedence of operations.
• Without parentheses, exponentiation typically occurs before multiplication or negation.
• Simplifying algebraic expressions requires attention to both these conventions and the properties of operations like powers.

By grasping the nuances of algebraic expressions, students can simplify and solve them more effectively, avoiding common pitfalls like misapplying negative signs.