Problem 54 - Fundamentals of Mathematics

Question

Find the value of \(\left(\frac{2}{3}\right)^{2}+\frac{8}{27}\).

Step-by-Step Solution

Answer

The value of the expression is \( \frac{20}{27} \).

Step 1: Calculate the Square of the Fraction

First, calculate the square of \( \frac{2}{3} \). Use the formula for squaring a fraction, \( \left(\frac{a}{b}\right)^2 = \frac{a^2}{b^2} \). Here, \( a = 2 \) and \( b = 3 \), so:\[\left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2} = \frac{4}{9}\].

Step 2: Simplify the Addition

Add \( \frac{4}{9} \) and \( \frac{8}{27} \). To do this, they need a common denominator. The least common multiple of 9 and 27 is 27. Convert \( \frac{4}{9} \) to \( \frac{12}{27} \):\[\frac{4}{9} = \frac{4 \times 3}{9 \times 3} = \frac{12}{27}\]Thus, the addition becomes:\[\frac{12}{27} + \frac{8}{27} = \frac{12 + 8}{27} = \frac{20}{27}\]

Step 3: Conclude the Calculation

We find that the value of the expression \( \left(\frac{2}{3}\right)^{2} + \frac{8}{27} \) is equal to \( \frac{20}{27} \).

Key Concepts

Squaring Fractions

When you square a fraction, you essentially multiply the fraction by itself. This means if you have a fraction like \( \frac{a}{b} \), squaring it results in \( \left(\frac{a}{b}\right)^2 = \frac{a \times a}{b \times b} = \frac{a^2}{b^2} \).
For example, squaring the fraction \( \frac{2}{3} \) involves:

• Squaring the numerator: \( 2^2 = 4 \)
• Squaring the denominator: \( 3^2 = 9 \)

Thus, \( \left(\frac{2}{3}\right)^2 = \frac{4}{9} \). Squaring helps address problems where powers are involved, simplifying calculations by working with familiar shapes of fractions.

Common Denominators

Finding a common denominator is essential when you want to add or subtract fractions. A common denominator is a common multiple of the denominators of the fractions involved. Once the denominators are the same, you can easily combine the fractions.
For example, when adding \( \frac{4}{9} \) and \( \frac{8}{27} \):

• The denominators are 9 and 27.
• A common denominator is 27, since it is a multiple of 9 and 27 (27 divides evenly by both).

From here, you adjust the fractions accordingly, which paves the way for straightforward addition of fractions.

Fraction Addition

Adding fractions necessitates that you first have a common denominator. Once achieved, you simply add the numerators and leave the denominator unchanged. This allows combining parts of a whole in a consistent and understandable manner.
Consider the example of adding \( \frac{12}{27} \) and \( \frac{8}{27} \):

• The numerators: 12 and 8, are added together \( 12 + 8 = 20 \).
• The denominator remains 27.

The sum of \( \frac{12}{27} + \frac{8}{27} \) becomes \( \frac{20}{27} \). Always ensure that fractions are converted to equivalent fractions with the common denominator before performing addition.

Least Common Multiple

The least common multiple (LCM) is the smallest number that is a multiple of two or more numbers. It plays a critical role in finding common denominators for fraction operations.
To find the LCM of two numbers like 9 and 27:

• List the multiples of each number.
• 9 has multiples: 9, 18, 27, 36, ...
• 27 has multiples: 27, 54, 81, ...

The smallest common multiple here is 27, making it the LCM. Using the LCM, you adjust fractions so they have this common denominator, facilitating easier addition or subtraction. In our example, making 9 into 27 involved multiplying by 3, so \( \frac{4}{9} \) became \( \frac{12}{27} \).