1. How did you determine the LCD or the Least Common Denominator of each set of rational expressions?
2. What are the uses of LCD in solving rational expressions?
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1. How did you determine the LCD or the Least Common Denominator of each set of rational expressions?
2. What are the uses of LCD in solving rational expressions?
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Answer:
1. To determine the Least Common Denominator (LCD) of a set of rational expressions you need to find the least common multiple (LCM) of the denominators. Here are the steps involved:
a. Factorize all the denominators into their prime factors.
b. Identify the highest power of each prime factor that appears in any denominator.
c. Multiply these highest powers of prime factors to find the LCD.
Let's consider an example. Suppose we have the rational expressions (1/4x (1/6x^2 and (1/3x^3 the denominators are 4x 6x^2 and 3x^3 respectively.
a. We can factorize each denominator as follows: 4x = 2^2 * x 6x^2 = 2 * 3 * x * x and 3x^3 = 3 * x * x * x.
b. The highest power of 2 is 2^2 the highest power of 3 is 3 and the highest power of x is x^3.
c. To find the LCD multiply the highest powers: LCD = 2^2 * 3 * x^3 = 12x^3.
2. The LCD is essential in solving rational expressions because it allows us to add or subtract fractions or perform any operations involving rational expressions. By finding the LCD and adjusting the denominators of the rational expressions to be the same we can combine them into a single fraction.
For example let's say we have the rational expressions: (1/2x) + (3/4x). To add these fractions we need a common denominator. The LCD in this case is simply the least common multiple of the denominators which is 4x. By adjusting the fractions we have: (2/4x) + (3/4x). Now with a common denominator we can add the fractions: (2 + 3)/(4x) = 5/(4x).
Similarly when solving rational expressions equations the LCD is crucial in getting rid of the fractions by multiplying both sides of the equation by the LCD. This simplifies the equation and makes it easier to solve.
In summary the LCD enables us to combine simplify and solve rational expressions by ensuring that they have a common denominator.
Answer:
1. To find the Least Common Denominator (LCD) of a set of rational expressions, you need to factor the denominators and identify the common factors. The LCD is the product of all the unique factors with their highest powers.
For example, if you have the rational expressions 1/(x^2 - 4), 1/(x - 2), and 1/(x + 2), you would factor the denominators as (x - 2)(x + 2) and (x - 2). The LCD would then be (x - 2)(x + 2).
2. The LCD is useful in solving rational expressions because it allows you to combine or add/subtract fractions with different denominators. By finding the LCD, you can rewrite the fractions with a common denominator and perform operations on them.
For example, if you have the rational expressions 1/(x - 2) and 1/(x + 2), you need to find the LCD, which is (x - 2)(x + 2). Then, you can rewrite the fractions with the LCD as the denominator and add them together. This allows you to simplify the expression and solve for x.