Problem 15 - Algebra 1: Concepts and Skills

Question

Find the least common denominator of the pair of rational expressions. $$ \frac{x+1}{15 x}, \frac{25}{18 x^{3}} $$

Step-by-Step Solution

Answer

The least common denominator of the two rational expressions is \(2*3^{2}*5*x^{3}\).

Step 1: Factorize denominators

Find the prime factors of the denominators. Prime factorization of \(15x\) gives \(3*5*x\) and for \(18x^{3}\) it gives \(2*3^{2}*x^{3}\).

Step 2: Identify common and unique factors

The common factors in both the denominators are \(3\) and \(x\). The unique factors are \(2, 5\) and \(x^{2}\), which appear in the denominators of the individual fractions.

Step 3: Calculate LCM

The least common multiple (LCM) can be found by multiplying the highest power of all prime factors. This gives the LCM as \(2*3^{2}*5*x^{3}\) which is also the Least Common Denominator (LCD).

Key Concepts

Rational Expressions

A rational expression is simply a fraction where both the numerator and the denominator are polynomials. Just as with whole numbers, fractions in algebra need a common denominator when adding or subtracting. When working with rational expressions, finding this common ground is essential to simplify or combine these expressions effectively.

• These expressions often appear in algebraic equations and are characterized by variables and constants.
• Just like regular fractions, you can reduce rational expressions by canceling out common factors in the numerator and the denominator.
• However, ensure the variables do not take on values that would make the denominator zero, as this would make the expression undefined.

Recognizing rational expressions and mastering them is key for solving complex algebraic problems, especially when working with least common denominators to simplify them.

Prime Factorization

Prime factorization is the process of breaking down a number into its basic building blocks (prime numbers). It's a critical skill in several areas of mathematics, including determining the least common denominator.

• Prime factors are numbers that can only be divided by 1 and themselves without leaving a remainder, such as 2, 3, 5, 7, and so on.
• By expressing a number as a product of primes, you can easily identify factors that contribute to both numbers or entire expressions.
• For example, to factorize the denominator 15x, you need the factors \(3\), \(5\), and \(x\). For 18x³, the factors are \(2\), \(3^2\), and \(x^3\).

Prime factorization is always the first step when finding the least common denominator, as it helps identify all necessary factors to consider.

Least Common Multiple (LCM)

The least common multiple (LCM) is the smallest number that is a multiple of two or more numbers. When dealing with rational expressions, the LCM of the denominators becomes the least common denominator (LCD), allowing you to perform operations like addition or subtraction.

• To calculate the LCM, consider the highest power of each prime factor present in any of the numbers. This ensures that the resulting number is a multiple of all given numbers.
• In our example, for denominators 15x and 18x³, the LCM is found by combining both common and unique factors: \(2\), \(3^2\), \(5\), and \(x^3\).
• The computed LCM is \(2 \times 3^2 \times 5 \times x^3\).

Realizing the LCM in this context helps combine and simplify rational expressions, making calculations straightforward and manageable.