Problem 35 - Prealgebra and Introductory Algebra

Question

Write the fractions in terms of the LCM of the denominators. $$\frac{9}{(x-1)^{2}}, \frac{6}{x(x-1)}$$

Step-by-Step Solution

Answer

The fractions in terms of the LCM are: \(\frac{9x}{x (x-1)^{2}}\) and \(\frac{6(x-1)}{x (x-1)^{2}}\)

Step 1: Identify the Denominators

First, identify the denominators of the two fractions. The denominators are \((x-1)^{2}\) and \(x(x-1)\). The first one is a square of \((x-1)\), and the second is a product of \(x\) and \((x-1)\).

Step 2: Calculate the LCM

As per the factors of each denominator, the least common multiple (LCM) would be the product of different factors. Here the LCM would be \(x(x-1)^{2}\).

Step 3: Write the Fractions in Terms of LCM

Now, Rewrite each of the fractions with the LCM as the denominator. To do this, multiply the numerator and the denominator of each fraction by the missing factors in the original denominator to get the LCM.\n\nThe first fraction becomes \(\frac{9x}{x (x-1)^{2}}\) and the second fraction becomes \(\frac{6(x-1)}{x (x-1)^{2}}\).

Key Concepts

Least Common Multiple (LCM)

Understanding the Least Common Multiple (LCM) is critical when working with fractions, especially when adding, subtracting, or comparing fractions. The LCM of two or more numbers is the smallest number that is a multiple of all the given numbers. For example, the LCM of 4 and 6 is 12, because 12 is the smallest number that is both a multiple of 4 (4x3) and 6 (6x2).

When working with algebraic fractions, which can have variables in the denominators, the LCM is found by identifying the unique factors and selecting the highest powers of those factors present in any of the denominators. In the exercise provided, the denominators \(x-1)^2\) and \(x(x-1))\) share the factor \(x-1)\), but the first is squared. Therefore, the LCM must include \(x\) and the square of \(x-1)\), making \(x(x-1)^2\) the LCM. This is crucial for combining fractions into a single expression or comparing their sizes.

Fraction Operations

Fraction operations include addition, subtraction, multiplication, and division. Before you can perform addition or subtraction with fractions, they must have the same denominator, which is where the LCM comes into play. By finding the LCM of the denominators, you can write equivalent fractions with a common denominator, simplifying the addition or subtraction process.

For the multiplication of fractions, you simply multiply the numerators together and the denominators together. Division of fractions involves flipping the second fraction (the divisor) and then multiplying. Regardless of the operation, simplification may be necessary to express the fraction in its simplest form. In our exercise, once the LCM is determined, adjusting each fraction to have the LCM as its new denominator allows them to be easily combined or compared.

Algebraic Fractions

Algebraic fractions behave much like numerical fractions but include variables in the numerator and/or the denominator. Simplifying algebraic fractions can involve factoring polynomials and finding common factors to reduce the fraction to its simplest form. Moreover, when adding or subtracting algebraic fractions, we seek a common denominator akin to numerical fractions.

In the given exercise, \(\frac{9}{(x-1)^{2}}\) and \(\frac{6}{x(x-1)}\) are algebraic fractions with different denominators. By determining the LCM of these denominators, we can express both fractions with a unified denominator, allowing further operations such as adding or combining like terms. Algebraic skills are necessary to identify the correct factors and perform the manipulations to arrive at the LCM-based denominators.