Problem 52 - Prealgebra

Question

Expand and simplify each of the following. $$\left(\frac{3}{8}\right)^{2}\left(\frac{4}{3}\right)^{2}$$

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{1}{4}\).

Step 1: Expand Each Term

First, let's expand the expression by calculating the square of each fraction independently. This means we square both the numerator and the denominator for each fraction.So, for the first fraction: \[\left( \frac{3}{8} \right)^2 = \frac{3^2}{8^2} = \frac{9}{64}.\]Similarly, for the second fraction: \[\left( \frac{4}{3} \right)^2 = \frac{4^2}{3^2} = \frac{16}{9}.\]

Step 2: Multiply the Expanded Terms

Now that we have the expanded terms, we can multiply them together. Remember to multiply the numerators together and the denominators together.The fractions to multiply are:\[\frac{9}{64} \text{ and } \frac{16}{9},\]so their multiplication is:\[\frac{9 \times 16}{64 \times 9} = \frac{144}{576}.\]

Step 3: Simplify the Resulting Fraction

Next, simplify the resulting fraction \( \frac{144}{576} \). To do this, find the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 144 and 576 is 144.Divide both the numerator and the denominator by this GCD:\[\frac{144 \div 144}{576 \div 144} = \frac{1}{4}.\]

Step 4: Conclusion

The expanded and simplified form of the original expression \(\left(\frac{3}{8}\right)^2\left(\frac{4}{3}\right)^2\) is \(\frac{1}{4}\).

Key Concepts

Expanding Expressions

When we talk about expanding expressions, we are referring to the process of taking a mathematical expression that is in a condensed form and rewriting it so that all of its components are clearly visible. In our exercise, we expanded expressions that involved squaring fractions, which means multiplying a fraction by itself. For example, the expression \( \left(\frac{3}{8}\right)^2 \) involves multiplying \( \frac{3}{8} \) by itself. This process is done by squaring both the numerator and the denominator separately. So, \( 3^2 = 9 \) and \( 8^2 = 64 \), making the expanded form \( \frac{9}{64} \).

• Expanding helps in visualizing what each part of the expression contributes to the whole.
• It's a crucial step before you simplify or perform further operations.

In general, expanding expressions is a foundational skill that makes handling complex expressions easier by making each component of the expression explicit.

Squaring Fractions

Squaring a fraction may sound tricky at first, but it’s actually quite straightforward once you know the steps. The term 'squaring' indicates multiplying a number by itself. So, when you square a fraction, you multiply the fraction by itself. In mathematical terms, if you have a fraction \( \frac{a}{b} \), squaring it involves calculating \( \left(\frac{a}{b}\right)^2 = \frac{a^2}{b^2} \).

• This involves calculating the squares of both the numerator \(a\) and the denominator \(b\).
• For instance, with \( \frac{4}{3} \), squaring results in \( \frac{16}{9} \) because \(4^2 = 16\) and \(3^2 = 9\).

Once each fraction is squared independently, you can proceed by multiplying them together, as seen in our exercise. The key advantage of knowing how to square fractions is that it allows you to work with expressions in a more manageable form and makes it easier to see how values interact with each other when combined in equations.

Greatest Common Divisor (GCD)

Finding the greatest common divisor (GCD) is a pivotal step when simplifying fractions. The GCD of two numbers is the largest number that divides both of them without leaving a remainder. This tool helps reduce fractions to their simplest form by dividing both the numerator and the denominator by their GCD, thus achieving the most simplified version of the fraction.

• For instance, when dealing with the fraction \( \frac{144}{576} \), the GCD is found to be 144.
• This means both 144 and 576 can be divided evenly by 144, hence, dividing by the GCD simplifies the fraction to \( \frac{1}{4} \).

It is an essential concept as it not only helps in simplifying fractions but also aids in understanding the relationship between numbers within an expression. Moreover, using the GCD helps in revealing the simplest form of an answer, which is often useful in comparing fractions or finding common denominators.