Problem 90 - Fundamentals of Mathematics

Question

For the following problems, find the products. Be sure to reduce. $$\left(\frac{2}{11}\right)^{2}$$

Step-by-Step Solution

Answer

The solution is \(\frac{4}{121}\).

Step 1: Understand the Problem

We need to find the product of \(\frac{2}{11}\) squared, which means multiplying \(\frac{2}{11}\) by itself.

Step 2: Multiplying the Fractions

To square the fraction \(\frac{2}{11}\), multiply the numerators and the denominators: \[\left(\frac{2}{11}\right) \times \left(\frac{2}{11}\right) = \frac{2 \times 2}{11 \times 11} = \frac{4}{121}\]

Step 3: Simplify the Result

Check the fraction \(\frac{4}{121}\) to see if it can be reduced. The greatest common divisor (GCD) of 4 and 121 is 1, so \(\frac{4}{121}\) is already in its simplest form.

Key Concepts

Fraction Simplification

Fraction simplification is the process of reducing fractions to their simplest form. While it's called 'simplification', this means finding an equivalent fraction with the smallest possible numerator and denominator.

Here's how to simplify a fraction:

• Identify the numerator (top number) and the denominator (bottom number) of the fraction.
• Find the greatest common divisor (GCD) of these two numbers. The GCD is the largest number that divides both the numerator and the denominator evenly.
• Divide both the numerator and the denominator of the fraction by their GCD. The result is the fraction's simplest form.

In the case of the fraction \(\frac{4}{121}\), the GCD is 1, as there are no common factors between 4 and 121 other than 1. Therefore, \(\frac{4}{121}\) is already simplified.

Squaring Fractions

Squaring fractions is a straightforward mathematical operation. It involves multiplying a fraction by itself to obtain a square of the original fraction.

When you square a fraction \(\frac{a}{b}\), you perform the following steps:

• Square the numerator: Multiply the numerator by itself, which means \(a \times a\) or \(a^2\).
• Square the denominator: Do the same for the denominator, which is \(b \times b\) or \(b^2\).
• Combine the squared terms to get \(\frac{a^2}{b^2}\).

In our exercise, we have the fraction \(\frac{2}{11}\). When squaring this fraction:

• Numerator becomes \(2 \times 2 = 4\).
• Denominator becomes \(11 \times 11 = 121\).

This results in \(\frac{4}{121}\), the square of the original fraction.

Greatest Common Divisor

The greatest common divisor (GCD) is an essential concept in simplifying fractions. It refers to the largest positive integer that divides two given numbers without leaving a remainder.

Finding the GCD is useful because it helps in reducing fractions to their simplest form.

Here's a simple way to find the GCD:

• List all divisors of both numbers.
• Identify the greatest common factor from these lists.
• The largest number common to both lists is the GCD.

For example, with numbers 4 and 121:

• The divisors of 4 are 1, 2, and 4.
• The divisors of 121 are 1 and 121.
• The only common divisor is 1, making it the GCD.

This is why the fraction \(\frac{4}{121}\) cannot be simplified further, as the GCD is 1.