Problem 72 - Algebra 1: Concepts and Skills

Question

Multiply the fractions. $$\frac{2}{5} \cdot \frac{2}{5} \cdot \frac{2}{5}$$

Step-by-Step Solution

Answer

The product of \(\frac{2}{5} \cdot \frac{2}{5} \cdot \frac{2}{5}\) is \(\frac{8}{125}\).

Step 1: Identify Fractions

First, identify the fractions that need to be multiplied. In this case, the fractions are \(\frac{2}{5}\), \(\frac{2}{5}\), and \(\frac{2}{5}\).

Step 2: Multiply Numerators

Multiply the numerators of the fractions together. Therefore, \(2 \times 2 \times 2 = 8\). This outcome will be the numerator of the final result.

Step 3: Multiply Denominators

Next, multiply the denominators of the fractions together. Hence, \(5 \times 5 \times 5 = 125\). The result will be the denominator of the answer.

Step 4: Write The Result

Lastly, combine the results of Step 2 and Step 3 to write the final answer. The answer is \(\frac{8}{125}\).

Key Concepts

Understanding the Numerator

The numerator is a vital part of a fraction, positioned on the top. It essentially tells you how many parts of a whole you are considering. When you multiply fractions, you focus on multiplying the numerators of the fractions involved.
In our example, each fraction has a numerator of 2. To find the overall numerator in the final result after multiplication, you perform the operation:

• Multiply the numerators together.
• For the fractions \(\frac{2}{5} \times \frac{2}{5} \times \frac{2}{5}\), we calculate \(2 \times 2 \times 2 = 8\).

This complete step is crucial, as the resulting numerator 8 becomes part of the final fraction \(\frac{8}{125}\). It shows the portion of the product compared to the whole that the newly multiplied fraction represents.

The Role of the Denominator

The denominator, located at the bottom of a fraction, reveals into how many parts the whole is divided. It remains consistent while you perform operations, maintaining the size of the fraction's sections.
To find the denominator when multiplying fractions, multiply the denominators of the fractions involved:

• For our fractions, each has a denominator of 5.
• Multiply them as follows: \(5 \times 5 \times 5 = 125\).

Ultimately, the denominator 125 is combined with the numerator to form the new fraction \(\frac{8}{125}\). This result indicates into how many segments the product is divided, remaining consistent with each original fraction.

Mastering Fraction Multiplication

Fraction multiplication might sound complex, but it is straightforward with a grasp on numerators and denominators. To multiply fractions, you need to:

• Multiply the numerators of the fractions to get the numerator of the product.
• Multiply the denominators to determine the denominator of the product.

To ensure clarity, take the fractions \(\frac{2}{5} \times \frac{2}{5} \times \frac{2}{5}\), and follow these steps:

• First, multiply the numerators: \(2 \times 2 \times 2 = 8\).


• Next, multiply the denominators: \(5 \times 5 \times 5 = 125\).


• Combine those results into a new fraction: \(\frac{8}{125}\).

This method outlines a simple yet powerful way to handle any fraction multiplication by ensuring the components combine correctly and logically.