Problem 10 - Beginning and Intermediate Algebra

Question

Evaluate. \(\left(\frac{6}{11}\right)^{2}\)

Step-by-Step Solution

Answer

The evaluated result is \( \frac{36}{121} \).

Step 1: Understand the Problem

We are asked to evaluate the expression \( \left(\frac{6}{11}\right)^{2} \), which means to calculate the square of the fraction \( \frac{6}{11} \).

Step 2: Square the Numerator

To square the fraction, start by squaring the numerator. The numerator is 6, so calculate \( 6^2 \). This equals 36.

Step 3: Square the Denominator

Next, square the denominator of the fraction. The denominator is 11, so calculate \( 11^2 \). This equals 121.

Step 4: Form the Fraction

Now that you have the squared numerator and denominator, form the new fraction. The result of squaring \( \frac{6}{11} \) is \( \frac{36}{121} \).

Step 5: Simplify the Fraction (if possible)

Check if the fraction \( \frac{36}{121} \) can be simplified. Since 36 and 121 have no common factors other than 1, the fraction is already in its simplest form.

Key Concepts

Fraction Multiplication

Multiplying fractions is a fundamental aspect of understanding fractional calculations. To multiply two fractions, you simply multiply the numerators together to get the new numerator, and the denominators together to get the new denominator. This is often seen in problems involving powers of fractions, like our exercise.

For example, when you see an expression like \( \left(\frac{a}{b}\right)^n \), it's the same as multiplying \( \frac{a}{b} \) by itself \( n \) times. In essence, you multiply the numbers in both the numerator and the denominator several times. This method retains the properties of the original fraction in its entirety, allowing for consistent and logical calculations.

Remember:

• Multiply numerators together.
• Multiply denominators together.

This method ensures that fractions are handled accurately, preserving the relationships between their parts.

Numerator and Denominator

The terms **numerator** and **denominator** are crucial in fractions. The numerator is the top part of a fraction, while the denominator is the bottom part. The numerator tells you how many parts you have, and the denominator tells you into how many equal parts the whole is divided.

Understanding their roles helps you grasp other concepts like squaring fractions. For instance, in the expression \( \left(\frac{6}{11}\right)^2 \), the number 6 is the numerator and the number 11 is the denominator.

When dealing with powers, as shown:

• **Squaring the numerator:** Square 6 to get 36.
• **Squaring the denominator:** Square 11 to get 121.

These operations individually result in a fraction where the outcome aligns with the original fraction's ratio or relationship. It's critical for accurate calculation and comparison purposes.

Simplification of Fractions

Simplifying fractions involves reducing them to their simplest form, where the numerator and denominator only share the factor of 1. This process makes fractions easier to understand and work with.

Consider our resulting fraction \( \frac{36}{121} \) from squaring \( \frac{6}{11} \). To simplify:

• Check for common factors in the numerator (36) and the denominator (121).
• If the greatest common factor is 1, it is already in its simplest form.

In this case, the numbers 36 and 121 do not share any common factors except for 1, so \( \frac{36}{121} \) is in its simplest form. Simplification might seem trivial, but it's a powerful tool that aids clarity in math problems, especially in reducing errors when dealing with more complex operations.