Problem 44 - Beginning and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understanding the Expression

The expression is \(-2^4\). It is important to note that the negative sign is not included in the exponentiation. This is equivalent to \(- (2^4)\). This indicates that only 2 is raised to the fourth power, and then the result is multiplied by -1.

Step 2: Calculate 2^4

Raise 2 to the power of 4. This means multiplying 2 by itself three more times: \(2 \times 2 \times 2 \times 2 = 16\).

Step 3: Apply the Negative Sign

Now that we have calculated \(2^4 = 16\), apply the negative sign from the original expression. Multiply the result by -1 to get \(-16\).

Key Concepts

Exponentiation

Exponentiation is a fundamental mathematical operation where a number (the base) is multiplied by itself a specific number of times (the exponent). In the expression \(2^4\), 2 is the base, and 4 is the exponent. This means you multiply 2 by itself four times: \(2 \times 2 \times 2 \times 2 = 16\). Exponentiation helps in simplifying large multiplications and is key in various applications like scientific notation, computing, and engineering.

• Base: The number that is multiplied by itself. In \(2^4\), the base is 2.
• Exponent: The power to which the base is raised. In \(2^4\), the exponent is 4.

Remember, when the exponent is positive, it indicates repeated multiplication. Understanding this process is crucial for solving powers and roots effectively.

Negative Numbers

Negative numbers are numbers that are less than zero and are typically denoted by a minus sign (-) in front. In mathematical expressions, negative numbers have rules that affect how expressions and calculations are carried out. For example, in the expression \(-2^4\), the negative sign applies to the entire expression after the exponentiation is evaluated. This means you first calculate \(2^4 = 16\), and then apply the negative sign to make it \(-16\).

• Negative Symbol: Indicates a number less than zero. Always applied after any operations such as multiplication or exponentiation.
• Order of Operations: When dealing with negative numbers in expressions, it's important to apply arithmetic operations in the correct order. Calculate powers or brackets first, then apply the negative sign.

Using negative numbers requires careful attention to their placement in expressions, especially with operations involving powers.

Mathematical Expressions

Mathematical expressions combine numbers, operators, and symbols to represent values and equations. Understanding how to correctly interpret and evaluate them is essential. The expression \(-2^4\) involves several steps that highlight the importance of following the order of operations.

• Components: Expressions consist of numbers, operators (like \(+,-,\times,\div\)), and often variables or constants.
• Order of Operations: This follows the BODMAS/BIDMAS rule—Brackets, Orders (powers and roots), Division and Multiplication (from left to right), Addition and Subtraction (from left to right).
• Evaluation Process: Evaluate powers before applying other operations, as shown in \(2^4\) being evaluated to 16, before applying the negative sign.

Knowing how to understand and solve mathematical expressions ensures accuracy and confidence in solving math problems.