Problem 34 - Algebra 1: Concepts and Skills

Question

Copy and complete the statement. $$ \left(\frac{-3}{5}\right)^{2}=\frac{?}{25} $$

Step-by-Step Solution

Answer

The completed statement is \(\left(\frac{-3}{5}\right)^{2}=\frac{9}{25}\).

Step 1: Identify Base and Exponent

The base here is \(\frac{-3}{5}\) and the exponent is '2'. An exponent of '2' indicates the number is being squared, or multiplied by itself.

Step 2: Apply Exponent to Numerator and Denominator Separately

Apply the exponent to both the numerator and the denominator separately to get: \((-3)^{2}\) and \(5^{2}\). The denominator \(5^{2}\) neatly equals '25', confirming it with the problem statement.

Step 3: Calculate Numerator

Now, calculate \((-3)^{2}\), which equals '9'.

Step 4: Verify Completed Statement

Thus, the completed statement is \(\left(\frac{-3}{5}\right)^{2}=\frac{9}{25}\).

Key Concepts

Fractions

Fractions are a way to represent numbers that are not whole. They consist of a numerator and a denominator. The numerator is the top number and indicates how many parts you have. The denominator, which is the bottom number, shows how many parts make a whole. For example, in the fraction \(\frac{3}{4}\), 3 is the numerator and 4 is the denominator.Fractions can be used in arithmetic just like whole numbers—added, subtracted, multiplied, or divided. When multiplying or dividing fractions, you handle the numerators and denominators separately. This is why they are a versatile tool in mathematics, allowing us to express everything from ratios to proportions.

Numerator and Denominator

Understanding the roles of the numerator and the denominator is crucial when working with fractions. The numerator is the number above the fraction line and indicates the count of equal parts being considered. The denominator, on the other hand, is the number below the line, indicating into how many total parts the whole is divided.

• Numerator: It captures the 'part' of the fraction. For example, in \(\frac{2}{5}\), the numerator is 2, indicating that we have 2 parts out of the total.
• Denominator: It captures the 'whole' of the fraction. In the same fraction \(\frac{2}{5}\), 5 is the denominator, indicating that the whole is split into 5 parts.

This balance between the numerator and denominator allows fractions to precisely describe values that lie between whole numbers.

Squaring Numbers

Squaring a number means multiplying that number by itself. This is an essential operation in mathematics, represented with an exponent of 2. When you square a number, you are essentially finding the area of a square whose sides are the length of the number you're squaring.For any fraction like \(\frac{a}{b}\), if you want to find \(\left(\frac{a}{b}\right)^2\), you simply square both the numerator and the denominator separately:

• Square the numerator: \(a^2\)
• Square the denominator: \(b^2\)

This method keeps the fraction intact and allows you to accurately calculate its square. For example, squaring \(\frac{-3}{5}\) involves calculating \((-3)^2\) and \(5^2\), resulting in the fraction \(\frac{9}{25}\). This ensures every part of the fraction is squared, maintaining proportionality.