Problem 151 - 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. $$(-5)^{3}=(-5)(-5)(-5)=?$$

Step-by-Step Solution

Answer

Therefore, \(-5^{3} = -125\)

Step 1: Understand exponents in multiplication

When a number has an exponent of 3, this indicates multiplication of the number by itself twice, meaning we must compute \(-5 \times -5 \times -5\) in this case.

Step 2: Compute the first multiplication

Let's perform the first multiplication. Two negatives make a positive when multiplied, so \(-5 \times -5 = 25\).

Step 3: Compute the final multiplication

Now, we multiply this result by \(-5\), which equals \(25 \times -5 = -125\).

Key Concepts

Understanding Repeated Multiplication

When you see an exponent, it means you will be performing repeated multiplication. For example, \((-5)^{3}\) tells us that \(-5\) is multiplied by itself a total of three times. Here's what it looks like in action:

• The expression \((-5)(-5)(-5)\) is our repeated multiplication.
• The exponent (3 in this case) tells us how many times to multiply the number by itself.

Repeated multiplication helps in simplifying expressions and understanding how numbers interact over multiple operations. When you get comfortable with breaking down exponents into repeated multiplication, it becomes easier to handle complex mathematical problems.
This concept is especially crucial when you start working with larger numbers or more complex equations.

Interpreting Negative Numbers in Multiplication

Negative numbers add a layer of complexity to multiplication problems. It's key to remember a fundamental rule:

• When you multiply two negative numbers, the result is a positive number.
• Multiplying a positive number by a negative number gives a negative result.

Let's apply this rule:

• The expression \((-5) \times (-5)\) results in \(25\) because multiplying two negatives gives a positive.
• However, when you multiply \(25 \times (-5)\), the result is \(-125\) because a positive times a negative equals a negative.

Understanding how negative numbers interact is essential in algebra and beyond. It allows you to handle different operations and signs accurately, without making common mistakes.

Evaluating Expressions with Exponents

Evaluating expressions with exponents involves breaking down the equation into simpler steps. This approach simplifies the process and improves accuracy.

• Start by understanding the exponent and translating it into repeated multiplication.
• Perform each multiplication step carefully, keeping track of positive and negative signs.

In the example \((-5)^{3} = (-5)(-5)(-5)\), we:

• Multiply the first two numbers: \((-5) \times (-5) = 25\).
• Then, multiply the result by the last number: \(25 \times (-5) = -125\).

By evaluating step by step, you ensure that each multiplication aligns with the rules of arithmetic. This method is not only useful for simple expressions but is also critical in more advanced math, ensuring your results are reliable and precise.