Problem 69 - Elementary and Intermediate Algebra

Question

Divide, if possible, and check. If a quotient is undefined, state this. $$ \frac{9.7(-2.8) 0}{4.3} $$

Step-by-Step Solution

Answer

The quotient is \(0\).

Step 1 - Multiplying the Numerators

First, multiply all the numbers in the numerator: \(9.7\), \(-2.8\), and \(0\).

Step 2 - Apply Zero Multiplication Property

Coming from Step 1, we notice that any number multiplied by zero equals zero. Therefore, \(9.7 \times -2.8 \times 0 = 0\).

Step 3 - Divide the Result by the Denominator

Now, take the result from Step 2, which is \(0\), and divide it by the denominator, \(4.3\). So, \( \frac{0}{4.3} = 0 \).

Step 4 - Final Answer

Since the division of zero by any non-zero number is zero, the final quotient is \(0\).

Key Concepts

zero multiplication property

In algebra, the zero multiplication property is extremely important. This rule states that any number multiplied by zero is zero. It doesn’t matter how big or how small the number is, if you multiply it by zero, the result is always zero.

Let’s look at the exercise to understand this property better. When you multiply 9.7, -2.8, and 0, you first multiply all the numbers together. But here’s the trick, you don’t even need to complete all those multiplications because as soon as you see the zero, you know the answer will be zero. This is exactly what happens in the given exercise:

• Multiply 9.7 and -2.8, then multiply that product by 0
• Since any number multiplied by 0 is 0, the entire numerator becomes 0

So, the zero multiplication property simplifies your work a lot and makes solving algebra problems much quicker when you identify a zero multiplier early on.

division by non-zero number

When you deal with division in algebra, understanding the role of zero and non-zero numbers is critical. In the exercise, after simplifying the numerator to zero, the next step is to divide by 4.3, which is a non-zero number.

This brings us to a key algebra fact: any number divided by a non-zero number is defined, and if the numerator is zero, the result is always zero. Here’s how that plays out:

• Take your simplified numerator, which is 0
• Now divide this by the denominator, which is 4.3
• Since 0 divided by any non-zero number is 0, your answer is 0

The important takeaway is understanding that division by a non-zero number is straightforward, and in our specific case, provides a clear result since 0 divided by 4.3 is 0.

numerator and denominator

To fully grasp any division problem in algebra, it's essential to understand the roles of the numerator and denominator. The numerator is the top part of the fraction and the denominator is the bottom part.

In the given exercise:

• Numerator: 9.7(-2.8)0
• Denominator: 4.3

Initially, we need to simplify the numerator. Once the numerator was simplified to zero (due to the zero multiplication property), we divided it by the denominator.

The rules are:

• Numerator ÷ Denominator = Quotient
• 0 ÷ Any Non-zero Number = 0

By applying these steps, we confirm that the numerator's zero leads to a final quotient of zero when divided by the non-zero denominator.