Problem 19 - College Algebra with Modeling and Visualization

Question

Exercises \(I 7-34:\) Evaluate the expression by hand. Check your result with a calculator. $$ -2^{4} $$

Step-by-Step Solution

Answer

The expression \(-2^4\) evaluates to \(-16\).

Step 1: Understand the expression

The given expression is \(-2^4\). It's important to note that the base in the expression is \(-2\), and the exponent is 4. The lack of parentheses indicates that only the 2 is being raised to the power of 4, not the negative sign.

Step 2: Apply the exponent

Calculate \(2^4\): \[2^4 = 2 \times 2 \times 2 \times 2 = 16\]. Since the negative sign is not included in the exponentiation, it remains outside of the calculation, so we have \(-2^4 = -(2^4) = -16\).

Step 3: Verify with a calculator

Use a calculator to evaluate \(-2^4\). Input the expression as -(2^4) to ensure the calculator interprets correctly. The calculator should confirm the result as \(-16\).

Key Concepts

Order of Operations

Order of operations is essential in mathematics to ensure consistent and accurate results when evaluating expressions. This rule is often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Understanding this sequence ensures we correctly solve mathematical expressions.
When dealing with exponentiation, it's crucial to recognize what is being raised to the power. In the expression \(-2^4\), under the typical order of operations, the exponent affects only the number immediately preceding it. Here, it is vital to understand that \-\ is not part of the base being raised to the fourth power. Consequently, we calculate \(-2^4\) as \(-1 \times 2^4\). This is a common area of confusion among students, so always pay attention to whether parentheses are present or not.

Negative Numbers

Negative numbers occur frequently in mathematics and it is important to understand how they interact with other numbers during operations like exponentiation. A negative number is any number that is less than zero, represented with a minus sign. In our expression, the negative sign appears before the base of the exponentiation, indicating that it is not part of the base.
It is crucial to understand that when no parentheses are involved, as in \(-2^4\), the negative sign is considered separate from the base. This differs from an expression like \((-2)^4\), where parentheses make the negative sign part of the base. In the latter case, the base is the entire number \(-2\), meaning both the 2 and the negative sign are raised to the fourth power, resulting in a positive number (16). However, without parentheses, the negative sign remains outside, giving us a negative result of \(-16\).

• Parentheses change how negative numbers interact with exponentiation.
• Always pay attention to whether the negative sign is included in the exponentiation or not.

Calculating Powers

When calculating powers, you are essentially multiplying a number by itself a specific number of times. This concept is vital for both large calculations and simplified expressions. In the expression \(2^4\), we multiply 2 by itself four times: \[2^4 = 2 \times 2 \times 2 \times 2 = 16\].
Calculating powers accurately requires careful multiplication steps. It helps to break down the multiplication to ensure clarity and avoid mistakes. For instance, start by multiplying the first two values \(2 \times 2 = 4\), then continue multiplying by 2 stepwise to complete the calculation \(4 \times 2 = 8\) and \(8 \times 2 = 16\).
Knowing how to calculate powers helps build a foundation for more advanced math topics, including polynomials and functions. Powers operate exponentially, meaning their growth increases rapidly, which is important to understand for mathematics and real-world applications. Be thorough with calculations to prevent errors, and confirm with a calculator when necessary.