Problem 76 - Introductory Algebra for College Students

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: Write down the given expression

The given expression is \(8 \div\left(-\frac{2}{9}\right)\).

Step 2: Convert Division to Multiplication

Convert the division operation into multiplication by changing the divisor \(-\frac{2}{9}\) to its reciprocal. The reciprocal of a fraction is created by swapping the numerator and denominator. Therefore, the reciprocal of \(-\frac{2}{9}\) is \(-\frac{9}{2}\). The expression now becomes: \(8 \times -\frac{9}{2}\).

Step 3: Perform the multiplication

Now, proceed with the multiplication operation. Multiply \(8\) by \(-\frac{9}{2}\) to obtain the final answer.

Key Concepts

Reciprocal

The term "reciprocal" might sound complex, but it's a concept that is quite easy to grasp. When we talk about the reciprocal of a number, we mean the number that, when multiplied with the original, gives a product of 1. This is especially common with fractions. To find the reciprocal of a fraction, simply swap its numerator and denominator.
Let's take an example. The fraction \(-\frac{2}{9}\) has a reciprocal \(-\frac{9}{2}\). Here, the negative sign remains in the reciprocal, often understood as part of the numerator. The multiplication of these two, \(-\frac{2}{9} \times -\frac{9}{2}\), results in 1, as the negatives cancel each other out and the fractions multiply through to get 1.

• The reciprocal of a whole number 'a' is \(\frac{1}{a}\).
• For a fraction 'a/b', the reciprocal is 'b/a'.
• A number and its reciprocal always multiply to make 1.

Expression Simplification

Simplifying expressions is like tidying up a messy house. It makes the math easier to deal with and often gives you the answer in a much clearer form. Consider simplifying the expression \(8 \div \left(-\frac{2}{9}\right)\). First, the division operation is turned into multiplication by using the reciprocal of \(-\frac{2}{9}\), which is \(-\frac{9}{2}\). So, the expression changes into \(8 \times -\frac{9}{2}\).

• Originally, you might have to deal with division, which many find tricky.
• Convert the division into multiplication for simplicity.
• The result is easier to see: multiply straight off.

When using multiplication, it's often easier to get the answer correctly and quickly than working directly with complicated division terms. This simplifies the entire operation.

Undefined Expression

Sometimes, we come across expressions in math that are 'undefined'. This occurs if you try to divide by zero, for instance. A division by zero does not produce a valid number and hence is termed as undefined.In our given problem, however, we don't face an undefined situation. The expression \(8 \div \left(-\frac{2}{9}\right)\) is perfectly well-defined. While division by a negative fraction might look unfamiliar, it's entirely valid. The outcome is a clear multiplication when using the reciprocal, leading to numeric results rather than undefined features.

• You cannot divide by zero because it leads to mathematical contradictions.
• Always check the divisor to make sure it's not zero to avoid undefined results.
• Expressions with valid non-zero divisors produce defined and sensible results.

Understanding what makes an expression undefined is crucial so you can identify and manage such cases accurately during problem-solving.