Problem 45 - Elementary and Intermediate Algebra

Question

Find the LCD of each pair of rational expressions. \(\frac{35}{3 a^{2} b}, \frac{23}{a^{2} b^{3}}\)

Step-by-Step Solution

Answer

The LCD is \(3a^2b^3\).

Step 1: Identify the Denominators

The denominators given are \(3a^2b\) and \(a^2b^3\). Look at the factors of these denominators.

Step 2: List Prime Factors

Write the prime factorization of each denominator: \(3a^2b = 3 \times a^2 \times b^1\) and \(a^2b^3 = 1 \times a^2 \times b^3\).

Step 3: Determine the Highest Power of Each Factor

Identify the highest powers of all factors present in both denominators: the factors are \(3\), \(a\), and \(b\). The highest powers are \(3^1\), \(a^2\), and \(b^3\).

Step 4: Multiply the Factors

Multiply the highest powers of each factor together: \(3^1 \times a^2 \times b^3 = 3a^2b^3\). The result is the least common denominator.

Key Concepts

Understanding Rational Expressions

A rational expression is similar to a fraction, except instead of just having numbers in its numerator and denominator, it contains polynomials. The numerator and the denominator are both algebraic expressions. Rational expressions often involve variables, which can take on various values. This variability allows these expressions to represent a wide range of mathematical situations.

Key points to remember:

• To simplify a rational expression, you look for common factors in the numerator and denominator and divide them out.
• Rational expressions can be added, subtracted, multiplied, and divided, just like fractions. However, when adding or subtracting, you must find a common denominator.
• Once simplified, a rational expression can be evaluated by substituting specific values for the variables, as long as the denominator does not turn into zero. Division by zero is undefined.

Exploring Prime Factorization

Prime factorization is breaking down a number into its most basic building blocks, which are prime numbers. For example, the number 12 can be expressed as the product of the primes 2 × 2 × 3. Similarly, in algebra, we can apply prime factorization to algebraic terms in a rational expression, breaking them down into their simplest multiplying components.

Here's how it works:

• Each term in an algebraic expression can be factored into its prime elements, including constants and variables.
• When dealing with variables, each variable's "power" or exponent becomes a factor. For instance, the term \(a^2\) is already in prime factor form, consisting of two factors of \(a\).
• Prime factorization helps in determining the least common denominator (LCD) by making it easier to compare and combine denominators of rational expressions.

Identifying the Greatest Power of Factors

When working with rational expressions, especially when trying to find the least common denominator (LCD), identifying the greatest power of each factor is crucial. The greatest power of a factor refers to the highest exponent of a factor that appears in any of the expressions you're comparing.

To determine this:

• Look at each factor in the denominators of the rational expressions.
• For each distinct factor, compare the exponents in the different denominators.
• Select the largest exponent for that factor. For instance, if comparing \(b^1\) and \(b^3\), the greatest power of \(b\) would be \(b^3\).

This method ensures that when you find the LCD, it can adequately accommodate the factors present in all terms, consolidating the rational expressions seamlessly into one common denominator without any loss of information or accuracy.