Problem 5 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The expression evaluates to -16.

Step 1: Understand the expression

The expression \[-2^4\]is given without parentheses around the -2. This means only 2 is raised to the fourth power, and then the result is negated.

Step 2: Calculate 2 to the power of 4

First, calculate the value of \[2^4\].This equals \[2 \times 2 \times 2 \times 2 = 16\].

Step 3: Apply the negative sign

Now that we know \[2^4 = 16\],we apply the negative sign from the expression \[-2^4\],resulting in \[-16\].

Key Concepts

Order of Operations

In mathematics, we must follow specific rules to ensure everyone solves problems in the same way. This is where the order of operations comes in. The order of operations is often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

When evaluating expressions, it is important to perform calculations in this correct order:

• Parentheses: Perform operations inside parentheses first.
• Exponents: Next, calculate powers and roots.
• Multiplication and Division: These are next, moving from left to right.
• Addition and Subtraction: Finally, handle these operations from left to right.

In the expression \(-2^4\), because there are no parentheses directly around the negative sign, we must first calculate the exponent, which is \(2^4 = 16\), and then apply the negative sign, resulting in \(-16\). By following the order of operations, we ensure accurate and consistent results.

Negative Numbers

Negative numbers are numbers less than zero. They can be thought of as the opposite of positive numbers. They are often used in mathematics to represent values below a reference point, like below sea level or below zero degrees.When working with expressions involving negative numbers, it is important to understand how they interact with other numbers, especially during operations like multiplication and exponentiation. In the expression \(-2^4\), the absence of parentheses around -2 means the exponent only applies to 2. After calculating \(2^4\) and getting 16, we then apply the negative sign, resulting in \(-16\). This shows that negative numbers, when applied after exponentiation, simply change the sign of the result.Negative numbers can sometimes feel tricky, but with practice and understanding, working with them becomes much easier. They are an essential part of nearly all areas of mathematics.

Evaluating Expressions

Evaluating expressions means finding the value of an expression by performing all indicated operations. This usually involves applying the order of operations to combine different numbers and operations correctly.

To evaluate more complicated expressions, it helps to break them down step by step:

• Identify: Break down the expression into smaller parts and identify any operations needed.
• Simplify: Use the order of operations to simplify the expression, step by step.
• Calculate: Perform any arithmetic calculations necessary.

In our exercise with \(-2^4\), understanding the operator precedence helped us see that no parentheses meant the exponent applies to the base 2 alone. After calculating that, we negated the result to find the value \(-16\). This methodical approach ensures precision and clarity.

Evaluating expressions is a foundational skill in mathematics that builds towards more complex problem-solving tasks. Practicing this skill helps in understanding and solving more advanced mathematical problems down the road.