Problem 50 - Prealgebra

Question

Evaluate \((x y)^{2}\) if \(x=\frac{3}{4}\) and \(y=-\frac{4}{5}\).

Step-by-Step Solution

Answer

The value of \((xy)^{2}\) is \(\frac{9}{25}\).

Step 1: Substitute the given values for x and y

We know the values of \(x\) and \(y\). The problem gives us \(x = \frac{3}{4}\) and \(y = -\frac{4}{5}\). First, substitute these values into the expression \((xy)^{2}\) to get \((\frac{3}{4} \times -\frac{4}{5})^{2}\).

Step 2: Compute the product inside the parentheses

Now, perform the multiplication \(\frac{3}{4} \times -\frac{4}{5}\). Multiply the numerators together: \(3 \times -4 = -12\) and the denominators together: \(4 \times 5 = 20\). This gives \(\frac{-12}{20}\).

Step 3: Simplify the fraction

The fraction \(\frac{-12}{20}\) can be simplified by finding the greatest common divisor (GCD) of 12 and 20, which is 4. Divide both the numerator and the denominator by 4 to get \(\frac{-3}{5}\).

Step 4: Calculate the square of the simplified fraction

Now, we need to square the simplified fraction \(\left(\frac{-3}{5}\right)^{2}\). Square the numerator: \((-3)^{2} = 9\) and the denominator: \(5^{2} = 25\). This gives us \(\frac{9}{25}\).

Key Concepts

Fractions

A fraction is a way to represent a part of a whole. It consists of two numbers: the numerator and the denominator. The numerator is the top number, and it indicates how many parts are being considered. The denominator is the bottom number, and it shows the total number of equal parts the whole is divided into. For example, in the fraction \(\frac{3}{4}\), 3 is the numerator and 4 is the denominator. This means we are considering 3 parts out of a total of 4.
Understanding fractions is crucial because they often appear in various areas of mathematics, such as ratios, proportions, and dividing quantities.

Multiplication

Multiplication in mathematics is one of the basic arithmetic operations. When dealing with fractions, the process involves multiplying the numerators together and the denominators together. This is different from adding or subtracting fractions, where finding a common denominator is necessary. For example, when multiplying \(\frac{3}{4}\) by \(\frac{-4}{5}\), you multiply the numerators (3 and -4) to get -12, and the denominators (4 and 5) to get 20. This results in the fraction \(\frac{-12}{20}\).
Multiplying fractions is useful when solving problems that involve scale, area, and probability.

Exponents

Exponents indicate how many times a number, known as the base, is multiplied by itself. In our context, squaring a fraction like \(\left(\frac{-3}{5}\right)^{2}\) means multiplying the fraction by itself once. To compute this, you square the numerator (-3) and the denominator (5) separately.

• Numerator: \((-3)^{2} = 9\)
• Denominator: \(5^{2} = 25\)

This results in \(\frac{9}{25}\). Understanding exponents is fundamental for working with powers, roots, and scientific notation.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator and the denominator have no common factors other than 1. This makes the fraction easier to understand and work with in calculations. To simplify \(\frac{-12}{20}\), we find the greatest common divisor (GCD) of 12 and 20, which is 4. We then divide both the numerator and the denominator by their GCD.

• Numerator: \(-12 \div 4 = -3\)
• Denominator: \(20 \div 4 = 5\)

This results in the simplified fraction \(\frac{-3}{5}\). Mastering simplification of fractions is essential in algebra, where expressions need to be made more manageable.