Problem 152 - Introductory and Intermediate Algebra for College Students 4th

Question

Exercises \(150-152\) will help you prepare for the material covered in the next section. In each exercise, an expression with an exponent is written as a repeated multiplication. Find this product, indicated by a question mark. $$(-2)^{4}=(-2)(-2)(-2)(-2)=?$$

Step-by-Step Solution

Answer

The product (-2)^4 equals 16.

Step 1: Understanding the Problem

We have (-2) to the power 4, which means -2 is multiplied by itself 4 times. So, we have (-2)(-2)(-2)(-2).

Step 2: Multiplying the First Pair

Firstly, multiply the first pair of -2's. That is, multiply (-2) * (-2) which equals 4.

Step 3: Multiplying the Second Pair

Next, multiply the second pair of -2's. This also equals 4.

Step 4: Calculating the Final Result

Finally, multiply the results from Step 2 and 3, i.e., multiply 4 * 4. The result is 16.

Key Concepts

repeated multiplication

Repeated multiplication is a core concept when dealing with exponents. In mathematics, repeated multiplication is the process of multiplying a number by itself several times. When numbers are expressed with exponents, they tell us how many times to use the base as a factor. This can save a lot of writing and is particularly useful for handling large powers.

For example, in the expression \((-2)^4\), the base is -2, and the exponent is 4, indicating that -2 should be multiplied by itself 4 times. So, \((-2)^4\) is equivalent to \((-2)\times(-2)\times(-2)\times(-2)\). This type of problem becomes more manageable when tackled step by step, by grouping and multiplying them in pairs as shown in the original solution.

powers

The term 'powers' is another way to understand the concept of exponents. Powers consist of two parts: the base and the exponent. The base is the number being multiplied. The exponent, on the other hand, tells us how many times to multiply the base by itself. This is often read as the base 'raised to the power' of the exponent.

Consider the expression \((-2)^4\). Here, -2 is the base, and 4 is the exponent or power. This notation simply means multiply -2 by itself four times. Calculating powers requires careful attention to signs, particularly when dealing with negative numbers. For even-powered bases like \((-2)^4\), all multiplied signs cancel out, resulting in a positive product, whereas odd powers would result in a negative product.

integer operations

Understanding integer operations is crucial for calculating expressions with exponents. When multiplying integers, the rules of positive and negative numbers apply. Specifically, the sign rules for multiplication are:

• Positive \(\times\) Positive = Positive
• Negative \(\times\) Negative = Positive
• Positive \(\times\) Negative = Negative
• Negative \(\times\) Positive = Negative

The product of two negative integers is a positive integer, as seen in the calculation \((-2)\times(-2) = 4\). When working with exponents like \((-2)^4\), we can break it down in steps, using these rules. By multiplying the negative pairs, we eventually achieve a positive product. This ensures that calculations are accurate and mistakes are minimized by applying correct integer operations at every multiplication step.