Problem 76 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated division or state that the expression is undefined. $$ 8 \div\left(-\frac{2}{9}\right) $$

Step-by-Step Solution

Answer

-36

Step 1: Identify the operation

The operation presented here is a division: 8 divided by -2/9.

Step 2: Reciprocate the second number

To divide by a fraction, we multiply by its reciprocal. The reciprocal of a fraction is obtained by switching its numerator and denominator. Therefore, the reciprocal of -2/9 is -9/2.

Step 3: Perform the multiplication

Now, replace division operation by multiplication operation. The expression changes to: 8 times the reciprocal of -2/9, which is -9/2. Multiply these two numbers to get the answer. In this case, 8 times -9/2 equals -36.

Key Concepts

Reciprocal of a Fraction

When dividing by a fraction, such as dividing 8 by \( -\frac{2}{9} \), the process involves using the reciprocal of the fraction you're dividing by. But what exactly is a reciprocal? The reciprocal of a fraction is simply the inverse of that fraction, where the numerator and denominator are reversed. For example, the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \), provided that neither \( a \) nor \( b \) is zero, as division by zero is undefined in mathematics.

For the case of \( -\frac{2}{9} \), its reciprocal is \( -\frac{9}{2} \). This concept can sometimes be tricky, especially when dealing with negative fractions. Remember that the negative sign can be associated with either the numerator or the denominator, and it still represents the same value. The key takeaway is that when you divide by a fraction, you actually multiply by its reciprocal, making the division much simpler to perform.

Performing Multiplication

Once we have the reciprocal of the fraction, we move on to performing multiplication. This step is straightforward but crucial. Continuing with our example, we've replaced the division with a multiplication: 8 times \( -\frac{9}{2} \).

To multiply these two, place them over a common denominator to make calculations simpler. Since one of the numbers (8) is a whole number, we can think of it as \( \frac{8}{1} \). Now, multiplying two fractions involves multiplying their numerators together and their denominators together. This means multiplying 8 by -9 and 1 by 2, resulting in \( \frac{8 \times -9}{1 \times 2} = \frac{-72}{2} \). Simplifying that, we get -36 as the final answer. This shows that division by a fraction can be converted into a multiplication problem, which is often easier to handle.

Undefined Mathematical Expressions

One of the fundamental rules in mathematics is that division by zero is not allowed; any expression that proposes this is said to be undefined. This is because dividing something by zero does not give a meaningful or finite value. In practical terms, if you have zero groups of a number, you cannot determine how many items were in each group since there were no groups to begin with.

An undefined expression doesn't have a solution within the set of real numbers. That's why, when we take the reciprocal of a fraction, it's important to ensure the original denominator isn't zero. If it were, flipping it to make it the numerator would result in division by zero when attempting to multiply. It's these subtleties in understanding which can prevent calculation errors and deepen comprehension of math as a consistent and logical system of rules.