Problem 47 - Algebra A Combined Function

Question

Perform the indicated operation. \(\frac{2}{3}\left(-\frac{4}{9}\right)\)

Step-by-Step Solution

Answer

The result of the operation is \( \frac{-8}{27} \).

Step 1: Understand the Operation

The expression \( \frac{2}{3} \left(-\frac{4}{9}\right) \) involves multiplication between the fraction \( \frac{2}{3} \) and the negative fraction \( -\frac{4}{9} \). The key is to multiply both numerators and both denominators, while keeping track of the sign.

Step 2: Multiply the Numerators

Multiply the numerators of the two fractions. The numerators are \( 2 \) and \( -4 \). Thus, \( 2 \times (-4) = -8 \).

Step 3: Multiply the Denominators

Multiply the denominators of the two fractions. The denominators are \( 3 \) and \( 9 \). Thus, \( 3 \times 9 = 27 \).

Step 4: Form the Product Fraction

Combine the results from Step 2 and Step 3 to form a single fraction. This gives us \( \frac{-8}{27} \).

Step 5: Check for Simplification

Check if the fraction \( \frac{-8}{27} \) can be simplified further. Since 8 and 27 have no common factors other than 1, the fraction is already in its simplest form.

Key Concepts

Negative Numbers

When dealing with fractions, encountering negative numbers can initially be confusing. However, they are quite straightforward once you get the hang of them. In the example \(-\frac{4}{9}\), this fraction indicates a negative number. A negative sign before a fraction means that the value of that fraction is less than zero.

Always keep these quick tips in mind when multiplying with negative numbers:

• Multiplying a positive number with a negative number results in a negative number.
• Multiplying two negative numbers gives a positive result.

For the exercise, you multiply \(2\) by \(-4\), which results in \(-8\), adhering to the rule of multiplying a positive with a negative to get a negative result.

Numerators

The numerator is the top part of a fraction and indicates how many parts of the whole are being considered. In our example, we have numerators \(2\) and \(-4\). To multiply fractions, you multiply their numerators.

• Simply take the numerators from each fraction.
• Multiply them together: \(2 \times (-4) = -8\).

By focusing on numerators during multiplication, you form the top part of your result fraction. Remember, the sign of the product will be determined by the rules of multiplying positive and negative numbers.

Denominators

The denominator is found below the fraction line and shows the total number of equal parts the whole is divided into. In the multiplication of \(\frac{2}{3}\) and \(-\frac{4}{9}\), the denominators are \(3\) and \(9\).

For multiplication of fractions:

• Simply multiply the two denominators: \(3 \times 9 = 27\).

This becomes the denominator of the final result fraction, indicating it still consists of 27 parts. The denominator affects only the size of the parts, not their sign.

Simplifying Fractions

Simplifying fractions involves reducing them to their smallest form, where the numerator and denominator are as small as possible but still have the same value. For this objective, you need the greatest common factor (GCF). In \(\frac{-8}{27}\), we check the GCF for 8 and 27, which is 1, implying there are no common factors to simplify further.

Steps to simplify:

• Look for common factors between the numerator and denominator.
• Divide both by the greatest common factor.

Since no simplification was possible in the example, \(-\frac{8}{27}\) is already in its simplest form. Simplifying helps with clarity and ensures your answer is easy to interpret.