Problem 16 - Prealgebra

Question

Find each of the following products. (Multiply.) $$\frac{1}{4} \cdot \frac{3}{4} \cdot \frac{3}{4}$$

Step-by-Step Solution

Answer

The product is \(\frac{9}{64}\).

Step 1: Multiply the Numerators

Begin by multiplying the numerators of the fractions. This means multiplying \(1\), \(3\), and \(3\) together. Thus, you calculate \(1 \times 3 \times 3 = 9\).

Step 2: Multiply the Denominators

Next, multiply the denominators similarly. Multiply \(4\), \(4\), and \(4\) together: \(4 \times 4 \times 4 = 64\).

Step 3: Form the Fraction

Combine the results from Steps 1 and 2 to form a new fraction. The product of the fractions is \(\frac{9}{64}\).

Step 4: Simplify the Fraction

Check if the fraction can be simplified. Since \(9\) and \(64\) have no common factors other than \(1\), \(\frac{9}{64}\) is already in its simplest form.

Key Concepts

Numerators as the Top Part of a Fraction

When you're working with fractions, the top part of a fraction is called the **numerator**. Think of it as a way to tell us how many parts we have. In the fraction \(\frac{1}{4}\), the number \(1\) is the numerator, representing one part out of four whole parts.
To multiply fractions, you multiply the numerators together. For example, in the problem \(\frac{1}{4} \cdot \frac{3}{4} \cdot \frac{3}{4}\), the numerators are 1, 3, and 3. By multiplying them together, you get:

• \(1 \times 3 \times 3 = 9\)

The result is the numerator of your new fraction. By keeping the steps simple like this, it becomes easier to follow and solve multiplication problems with fractions.

Denominators Explained as the Bottom Part of a Fraction

A fraction's **denominator** is the bottom number, telling us into how many equal parts the whole is divided. For instance, in \(\frac{3}{4}\), the denominator is \(4\), showing that the whole is divided into four equal sections.
When multiplying fractions, you need to multiply the denominators. Continuing with our exercise, we have three denominators: 4, 4, and 4. Multiply them as follows:

• \(4 \times 4 \times 4 = 64\)

After calculating, \(64\) becomes the denominator in the resulting fraction. It's crucial to keep denominators accurately multiplied, so your final answer correctly represents the combined parts of each fraction involved.

The Art of Simplifying Fractions

**Simplifying fractions** means making a fraction as simple as possible. This involves reducing the fraction to its smallest form where the numerator and denominator have no common factors other than 1.
Once you have both the new numerator and denominator, as with \(\frac{9}{64}\) from earlier, check for common factors. In our example:

• The greatest common factor of 9 and 64 is 1.

Since they have no other common factors, \(\frac{9}{64}\) is already in its simplest form. Simplifying makes the fraction easier to understand and work with in further calculations or comparisons, ensuring efficiency and clarity in math problems.