Problem 96 - Fundamentals of Mathematics

Question

For the following problems, find the products. Be sure to reduce. $$\left(\frac{1}{2}\right)^{2} \cdot\left(\frac{2}{5}\right)^{2}$$

Step-by-Step Solution

Answer

The product is \( \frac{1}{25} \).

Step 1: Find Powers

First, calculate the powers of the fractions separately. Raise each fraction to its given exponent. For \( \left(\frac{1}{2}\right)^2 \), calculate: \( \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \). For \( \left(\frac{2}{5}\right)^2 \), calculate: \( \frac{2}{5} \times \frac{2}{5} = \frac{4}{25} \).

Step 2: Multiply the Fractions

Next, multiply the results of the powers together. Multiply the numerators: \(1 \times 4 = 4\) and the denominators: \(4 \times 25 = 100\). Therefore, \( \frac{1}{4} \cdot \frac{4}{25} = \frac{4}{100} \).

Step 3: Simplify the Fraction

Finally, simplify the fraction \( \frac{4}{100} \). Find the greatest common divisor of 4 and 100, which is 4. Divide the numerator and the denominator by their greatest common divisor: \( \frac{4 \div 4}{100 \div 4} = \frac{1}{25} \).

Key Concepts

Understanding Exponents in Fractions

When you encounter exponents in fractions, you're being asked to multiply the fraction by itself the specified number of times. For the fraction \( \left( \frac{1}{2} \right)^2 \), the exponent \( 2 \) means you should multiply the fraction \( \frac{1}{2} \) by itself:

• First, multiply the numerator: \( 1 \times 1 = 1 \).
• Next, multiply the denominator: \( 2 \times 2 = 4 \).

Therefore, \( \left( \frac{1}{2} \right)^2 = \frac{1}{4} \).
Similarly, for \( \left( \frac{2}{5} \right)^2 \), you do the same:

• Numerator: \( 2 \times 2 = 4 \).
• Denominator: \( 5 \times 5 = 25 \).

So, \( \left( \frac{2}{5} \right)^2 = \frac{4}{25} \).
By understanding how to handle exponents in fractions, you can easily convert them into their equivalent products before performing further operations.

Simplifying Fractions

After doing the multiplication of fractions, we often need to simplify the result to its smallest form. Simplification makes a fraction easier to read and work with. To simplify a fraction, like \( \frac{4}{100} \), you determine the greatest common divisor of the numerator and the denominator.
Once you have the greatest common divisor, you divide both the numerator and the denominator by this number. In our example:

• The greatest common divisor of \( 4 \) and \( 100 \) is \( 4 \).
• Divide the numerator \( 4 \) by \( 4 \) to get \( 1 \).
• Divide the denominator \( 100 \) by \( 4 \) to get \( 25 \).

Thus, \( \frac{4}{100} \) simplifies to \( \frac{1}{25} \). Simplifying fractions is fundamental in fraction mathematics as it allows you to express the result in its most refined form.

Finding the Greatest Common Divisor

The greatest common divisor (GCD) is a crucial concept when working with fractions, especially for simplifying. The GCD of two numbers is the largest number that can divide both without leaving a remainder.
To find the GCD, you can list the factors of each number and determine the largest common one. Alternatively, you can use the Euclidean algorithm, which involves repeated division. However, for smaller numbers, simply listing factors is often more straightforward.
Consider \( 4 \) and \( 100 \):

• Factors of \( 4 \): \( 1, 2, 4 \)
• Factors of \( 100 \): \( 1, 2, 4, 5, 10, 20, 25, 50, 100 \)

The largest number common to both lists is \( 4 \), making it their GCD. Using the GCD to simplify fractions helps ensure the fraction is in its most concise form, which is essential for accurate calculations and comparisons.