Problem 20 - College Algebra with Modeling and Visualization

Question

Exercises \(17-34:\) 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 expression \[ (-2)^4 \] asks us to evaluate the fourth power of -2.

Step 2: Calculate the Power

Raising a number to the fourth power means multiplying the number by itself four times. Thus, we need to calculate:\[ (-2) imes (-2) imes (-2) imes (-2) \]

Step 3: Multiply in Pairs

First, multiply the first two terms:\[ (-2) imes (-2) = 4 \]Next, multiply the result by the third term:\[ 4 imes (-2) = -8 \]Finally, multiply this result by the fourth term:\[ -8 imes (-2) = 16 \]

Step 4: Confirm the Result

The calculations show that \[ (-2)^4 = 16 \].Let's confirm with a calculator to ensure there are no errors. Inputting the expression \[ (-2)^4 \] into a calculator should yield the result 16.

Key Concepts

Arithmetic Operations

Understanding arithmetic operations is essential when working with exponential expressions. In the expression \((-2)^4\), arithmetic operations involve basic math functions like multiplication. You multiply numbers to find the product, and when dealing with exponents, it forms a repetitive multiplication.

Arithmetic operations help us break complicated calculations into simpler parts. You can think of \((-2)^4\) as doing

• \((-2) \times (-2)\) = 4, since two negative numbers multiply to a positive
• The next step is \(4 \times (-2)\), which results in -8
• Finally, multiplying \(-8 \times (-2)\) results in 16
  .

By working through it step-by-step with arithmetic operations, we can handle any similar problem without a calculator. Non-calculator methods also help improve mental math skills.

Power Calculation

Power calculation is a crucial part of evaluating exponential expressions. When you see a number raised to a power, like \((-2)^4\), the number inside the parentheses is the base, and the number outside is the exponent. The exponent indicates how many times we multiply the base by itself.

In this expression, \( 4 \) tells us to multiply \(-2\) four times, like this: \[ (-2) \times (-2) \times (-2) \times (-2) \]

The power of the base dictates repeated multiplication, helping to simplify expressions and find solutions quickly. This calculation style is essential for solving problems in algebra and beyond.

To solve the problem correctly, remember:

• The base is the number you multiply
• The exponent tells you how many times to multiply it
• Take it step-by-step, multiplying groups of pairs for simpler calculations

Negative Numbers

Handling negative numbers correctly is vital in expressions like \((-2)^4\). When dealing with negative bases, it is important to pay attention to the parity of the exponent.
An even exponent results in a positive product because of canceling out negative signs.

• Multiplying two negatives gives a positive
• With four multiplications, each pair of negatives results in a positive, retaining an overall positive result.

Understanding this concept helps avoid common mistakes in calculations. If the exponent were odd, the final result would remain negative due to the odd number of negative multiplications. With expressions similar to \((-2)^4\), knowing how negative numbers behave in sequences of multiplication helps solve them accurately. It's crucial to focus on these properties to excel in math.