Problem 43 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The expression \( \left(-\frac{3}{4}\right)^{2} \) simplifies to \( \frac{9}{16} \).

Step 1: Understand the Expression

The expression given is \( \left(-\frac{3}{4}\right)^{2} \). This means we need to raise the fraction \( -\frac{3}{4} \) to the power of 2.

Step 2: Apply the Power Rule to Fractions

When raising a fraction to a power, both the numerator and the denominator should be raised to that power. Therefore, \( \left(-\frac{3}{4}\right)^{2} \) becomes \( \frac{(-3)^{2}}{4^{2}} \).

Step 3: Compute the Squared Numerator

Calculate \( (-3)^{2} \). The square of \(-3\) is \( 9 \) since \( (-3) \times (-3) = 9 \).

Step 4: Compute the Squared Denominator

Calculate \( 4^{2} \). The square of \( 4 \) is \( 16 \) because \( 4 \times 4 = 16 \).

Step 5: Write the Simplified Expression

Combine the computed values from the numerator and the denominator. So, \( \frac{(-3)^{2}}{4^{2}} = \frac{9}{16} \).

Step 6: Final Verification

Double-check all calculations to verify correctness. The simplified expression of \( \left(-\frac{3}{4}\right)^{2} \) is indeed \( \frac{9}{16} \).

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential part of understanding mathematical expressions, especially when dealing with exponents. When you encounter a fraction, simplification is all about making the fraction as simple as possible without changing its value.
This process involves reducing the numerator and the denominator by their greatest common divisor (GCD). For example, if we have the fraction \( \frac{12}{16} \), we can simplify it by dividing both the numerator (12) and the denominator (16) by their GCD, which is 4. Therefore, \( \frac{12}{16} \) simplifies to \( \frac{3}{4} \).
It's important to remember that simplifying doesn't only apply before operations. Once calculations are done, you might also need to simplify the result. Always check if the final fraction can be reduced further, for better clarity and ease in understanding.

Numerator and Denominator

In a fraction, understanding the roles of the numerator and the denominator is crucial. The fraction \( \frac{a}{b} \) includes two main parts:

• Numerator - The number on the top of the fraction. It represents how many parts of a whole are being considered.
• Denominator - The number on the bottom of the fraction. It indicates how many equal parts the whole is divided into.

When working with exponents, both parts behave differently based on their roles. Raising a fraction to a power, for instance, requires both the numerator and the denominator to be treated independently but similarly.
For \( \left(-\frac{3}{4}\right)^{2} \), the numerator is \(-3\) and the denominator is \(4\). When squared, it becomes \( \frac{(-3)^2}{4^2} \), resulting in \( \frac{9}{16} \). Understanding how each component changes when squared or subjected to other operations is key to solving problems correctly.

Power Rule for Exponents

The power rule for exponents is a fundamental principle when working with expressions involving powers. It states that when you raise a base that is a fraction to a power, you must apply the power separately to both the numerator and the denominator.
For any fraction \( \left(\frac{a}{b}\right)^{n} \), the power rule converts it into \( \frac{a^{n}}{b^{n}} \). This ensures every part of the fraction is correctly scaled according to the power specified.
In the example \( \left(-\frac{3}{4}\right)^{2} \), the power rule guides us to compute \( (-3)^2 \) as well as \( 4^2 \), leading to \( \frac{9}{16} \).
This rule simplifies our calculation process, making complex expressions more manageable by breaking them down into simpler, familiar operations. Understanding this rule is paramount because it is employed often across various mathematical contexts, from algebra to calculus.