Problem 81 - Elementary Algebra Within Reach

Question

In Exercises \(75-86\), simplify the expression. $$ \left(\frac{2 x}{5}\right)\left(\frac{4 x}{8}\right) $$

Step-by-Step Solution

Answer

The simplified form of \( \left(\frac{2x}{5}\right)\left(\frac{4x}{8}\right) \) is \( \frac{x^2}{5} \).

Step 1: Multiply the Numerators

Firstly, you multiply the numerators of the fractions to create the numerator for your answer. Here, it would be \(2x \times 4x\) which gives us \(8x^2\).

Step 2: Multiply the Denominators

Next, you multiply the denominators of the fractions to create the denominator for your answer. In this exercise, that's \(5 \times 8\), which gives you 40.

Step 3: Form the Fraction

Having obtained the numerator and denominator, the fraction is formed by placing the numerator from Step 1 over the denominator from Step 2 thus obtaining \( \frac{8x^2}{40} \).

Step 4: Simplify the Fraction

The last step is to simplify the fraction. The numerator and the denominator have a common factor, which is 8, therefore there is a need to divide both by 8. This leads to \( \frac{x^2}{5} \).

Key Concepts

Fractions in Algebra

Understanding fractions in algebra is crucial for simplifying and manipulating algebraic expressions. In algebra, fractions consist of a numerator and a denominator, just like arithmetic fractions. However, sometimes variables accompany the numbers in these fractions, represented by letters like \(x\) or \(y\). For instance, in the expression \( \frac{2x}{5} \), \(2x\) is the numerator and \(5\) is the denominator. When dealing with algebraic fractions, the operations you perform, such as addition, subtraction, multiplication, or division, should consider both the numerical and the variable parts. Since the fractions contain variables, this can influence their behavior during operations. Simplifying fractions requires recognizing when you can cancel out common factors, especially when they include variables. Fractions in algebra are often used to express ratios, proportions, and other relationships in problems.

Multiplying Fractions

Multiplying fractions, whether numeric or algebraic, follows the same basic principle. Both the numerators and denominators are multiplied together separately.
For example, consider the fractions \( \frac{a}{b} \) and \( \frac{c}{d} \). When multiplying these fractions, you would calculate:

• Numerator: \( a \times c \)
• Denominator: \( b \times d \)

Once calculated, the result is the new fraction \( \frac{ac}{bd} \). With algebraic fractions, such as \( \left(\frac{2x}{5}\right)\left(\frac{4x}{8}\right)\), the same steps apply. Multiply the numerators: \(2x \times 4x\), resulting in \(8x^2\).
Then multiply the denominators: \(5 \times 8\), equaling \(40\). Hence, the product of these fractions is \( \frac{8x^2}{40} \). Remember to simplify at the end, which often involves factoring out any common factors.

Common Factors in Fractions

Simplifying fractions involves reducing them to their simplest form by removing any common factors between the numerator and denominator. To simplify a fraction like \( \frac{8x^2}{40} \), the first step is to identify any common factors both numbers share. Here, both 8 in the numerator and 40 in the denominator are divisible by 8.
Divide both numerator and denominator by their greatest common factor (GCF):

• Numerator: \( \frac{8x^2}{8} = x^2 \)
• Denominator: \( \frac{40}{8} = 5 \)

Thus, the simplified fraction becomes \( \frac{x^2}{5} \). Recognizing and dividing by common factors not only simplifies expressions but also makes them more manageable for further calculations or when solving equations. It's essential always to check for the greatest common factor to ensure the fraction is reduced adequately.