Problem 12 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

\( \frac{36}{121} \)

Step 1: Write down the expression

The given expression to evaluate is \( \left(\frac{6}{11}\right)^{2} \). This indicates that we need to square the fraction \( \frac{6}{11} \).

Step 2: Square the numerator and denominator

To square the fraction \( \frac{6}{11} \), we need to square both the numerator and the denominator separately. The expression becomes \( \frac{6^{2}}{11^{2}} \).

Step 3: Calculate the squares

Now, calculate the square of the numerator and the denominator: \( 6^{2} = 36 \) and \( 11^{2} = 121 \). Incorporating these results, the expression is \( \frac{36}{121} \).

Step 4: Simplify the fraction

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

Key Concepts

Understanding the Numerator

In the world of fractions, the term "numerator" refers to the top part of a fraction. It indicates how many parts of a whole are being considered. For instance, in the fraction \( \frac{6}{11} \), the number 6 is the numerator. This means we are focusing on 6 parts out of a total indicated by the denominator (which here is 11).

When dealing with fractional exponents, such as \( \left( \frac{6}{11} \right)^2 \), the numerator gets squared just like any integer or whole number would. Therefore, \( 6^2 \) equals 36.

To square a number, simply multiply the number by itself. Here are some quick steps:

• Identify the numerator of the fraction.
• Multiply this number by itself: \( n \times n \).
• Read the result, which becomes the new numerator of your squared fraction.

Exploring the Denominator

The denominator in a fraction is the number beneath the line. It tells us into how many equal parts the whole is divided. In \( \frac{6}{11} \), the denominator is 11, meaning the whole is broken into 11 equal parts.

When squaring a fraction, the denominator must also be raised to the power of two. For \( \frac{6}{11}^2 \), this becomes \( \frac{6^2}{11^2} \), where \( 11^2 \) equals 121.

To square a denominator:

• Locate the denominator value.
• Multiply this number by itself (e.g., \( d \times d \)).
• The outcome becomes the denominator of the squared fraction.

Squaring both the numerator and the denominator keeps the proportion the same, just changes the size of the parts being considered.

Simplifying Fractions

Once you have calculated the squares of both the numerator and the denominator in a fraction, the next step is to check if your fraction can be simplified. Simplifying a fraction means reducing it to its simplest form.

The fraction \( \frac{36}{121} \) is in simplest form if the greatest common divisor (GCD) of both 36 and 121 is 1. If they had any common factors other than 1, you would divide both by that factor repeatedly until no further simplification is possible.

Here's how typically you go about simplifying a fraction:

• List the factors of the numerator and the denominator.
• Determine the greatest common factor (GCF).
• Divide both the numerator and the denominator by the GCF.

In this exercise, since 36 and 121 have no common factors besides 1, \( \frac{36}{121} \) is already in its simplified form, indicating no further actions are required.