Problem 27 - Algebra A Combined Function

Question

Evaluate each expression. $$ \left(\frac{2}{3}\right)^{2} $$

Step-by-Step Solution

Answer

\( \frac{4}{9} \)

Step 1: Identify the Expression

We are given the expression \( \left(\frac{2}{3}\right)^{2} \). This expression indicates that we need to square the fraction \( \frac{2}{3} \).

Step 2: Apply the Squaring Rule

When you square a fraction, you square the numerator and the denominator separately. So, we need to calculate \( \left(\frac{2}{3}\right)^{2} = \frac{2^2}{3^2} \).

Step 3: Calculate Numerator and Denominator

Calculate \( 2^2 \) which is \( 4 \), and \( 3^2 \) which is \( 9 \). So we get \( \frac{4}{9} \).

Step 4: Write the Final Result

Combine the results from the previous step, and thus \( \left(\frac{2}{3}\right)^{2} = \frac{4}{9} \).

Key Concepts

Fractions

Fractions are a way to represent parts of a whole. They consist of two numbers separated by a line. The number above the line is the numerator, and the number below the line is the denominator. Fractions are essential for comparing parts of a whole, understanding proportions, and performing arithmetic operations with numbers less than one. They can also represent ratios and divisions. To be visually clear, let's see how fractions work:

• If you have a pizza, and you cut it into 4 equal slices, having one slice is a fraction of the whole pizza, specifically \( \frac{1}{4} \).
• The numerator \(1\) tells you how many parts you have, and the denominator \(4\) tells you into how many parts the whole is divided.

Understanding fractions is foundational for further learning in mathematics. You will see them in percentages, decimal conversions, and more.

Squaring Numbers

Squaring a number means multiplying the number by itself. This process increases the number significantly, especially if the original number is large.Squaring fractions follows a simple rule: square the numerator and the denominator separately. When you square \( \frac{2}{3} \), you apply the squaring rule to get \( \frac{2^2}{3^2} \). Here's why squaring should be a quick thing to grasp:

• For numbers: \( 3^2 \) is the same as \( 3 \times 3 = 9 \).
• For fractions: the same rule applies, just separated. So, \( \frac{2}{3}^2 \) becomes \( \frac{4}{9} \) after squaring each part.

This simple rule can have profound implications in geometry and algebra, where squaring adds dimensions to forms or equations.

Numerator and Denominator

Understanding the numerator and denominator is crucial when dealing with fractions. The numerator is the top number in a fraction and represents how many parts you are considering. The denominator is the bottom number, showing the total number of equal parts the whole is divided into.In the context of squaring fractions as in \( \left(\frac{2}{3}\right)^{2} \):

• Numerator: The number 2 becomes \( 2^2 = 4 \).
• Denominator: The number 3 becomes \( 3^2 = 9 \).

Square each independently, and you'll see these values then combine to form the squared fraction \( \frac{4}{9} \).By recognizing the distinct roles of the numerator and denominator, you refine your ability to manipulate fractions quickly and accurately in various mathematical contexts.