Problem 150 - 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. $$(-6)^{2}=(-6)(-6)=?$$

Step-by-Step Solution

Answer

The product is \(36\).

Step 1: Identify the Base and the Exponent

In this case, the base number is \(-6\), and the power or exponent is \(2\). This means we need to multiply \(-6\) by itself twice.

Step 2: Perform the Multiplication

Multiply \(-6\) by \(-6\) to get the product.

Step 3: Write the Answer

The product of \(-6\) And \(-6\) is \(36\), because the multiplication of two negative numbers yields a positive number.

Key Concepts

Understanding Negative Numbers Multiplication

When you multiply two negative numbers, you end up with a positive number. This happens because of the rules of negative numbers in multiplication. Here's why:
- A negative times a positive results in a negative.
- A negative times a negative results in a positive.
The signs "cancel out," leaving you with a positive number. In the example, \[ (-6) \times (-6) = 36 \]
So, the two negatives become a positive product.
Whenever you see two negative signs being multiplied, remember the result will be positive!

Base and Exponent Identification Made Simple

In expressions like \((-6)^2\), there's a base and an exponent.- The **base** is the number you're going to multiply. Here, it's \(-6\).- The **exponent** tells you how many times to multiply the base by itself. For this problem, the exponent is 2.Think of the exponent as telling you how many copies of the base you have. If the expression is \((-6)^2\), you're multiplying \(-6\) times itself, which appears twice.Knowing how to identify these parts helps in simplifying expressions and solving problems more efficiently.

The Magic of Repeated Multiplication

Exponents make repeated multiplication easier and more organized.For example, \((-6)^2\) is saying to multiply -6 by itself, just like writing \((-6) \times (-6)\).
This short form helps reduce the complexity of constant multiplication.- **Why use exponents?**
- They simplify expressions.
- They're easier to read and calculate, especially with larger bases or higher exponents.
Knowing that exponents simplify repeated multiplication gives you the power to handle complex calculations with ease!