What is the least common denominator of the rational expressions.
$\frac{-15}{10 x^3}, \frac{1}{8 x^2(x-6)^3}, \frac{14}{x^5\left(x^2-36\right)}$
A. $80 \cdot x^2 \cdot(x-6) \cdot(x+6)$
B. $40 \cdot x^2 \cdot(x-6) \cdot(x+6)$
C. $80 \cdot x^5 \cdot(x-6)^3 \cdot(x+6)$
D. $40 \cdot x^5 \cdot(x-6)^3 \cdot(x+6)$
Answer (1)
Find the least common multiple (LCM) of the coefficients: LCM(10, 8) = 40.
Identify the highest power of x: x 5 .
Identify the highest power of (x-6): ( x − 6 ) 3 .
Identify the highest power of (x+6): ( x + 6 ) . The least common denominator is 40 ⋅ x 5 ⋅ ( x − 6 ) 3 ⋅ ( x + 6 ) .
Explanation
Understanding the Problem We are given three rational expressions: 10 x 3 − 15 , 8 x 2 ( x − 6 ) 3 1 , x 5 ( x 2 − 36 ) 14 . We need to find the least common denominator (LCD) of these expressions.
Identifying Denominators The denominator of the first expression is 10 x 3 . The denominator of the second expression is 8 x 2 ( x − 6 ) 3 . The denominator of the third expression is x 5 ( x 2 − 36 ) .
Factoring the Third Denominator We can factor x 2 − 36 as ( x − 6 ) ( x + 6 ) . So the denominator of the third expression is x 5 ( x − 6 ) ( x + 6 ) .
Finding LCM of Coefficients Now we need to find the least common multiple (LCM) of the denominators. First, let's find the LCM of the coefficients: 10 and 8. The prime factorization of 10 is 2 × 5 , and the prime factorization of 8 is 2 3 . The LCM of 10 and 8 is 2 3 × 5 = 40 .
Finding Highest Power of x Next, we find the highest power of x in the denominators: x 3 , x 2 , x 5 . The highest power is x 5 .
Finding Highest Power of (x-6) Then, we find the highest power of ( x − 6 ) in the denominators: ( x − 6 ) 0 , ( x − 6 ) 3 , ( x − 6 ) 1 . The highest power is ( x − 6 ) 3 .
Finding Highest Power of (x+6) Finally, we find the highest power of ( x + 6 ) in the denominators: ( x + 6 ) 0 , ( x + 6 ) 0 , ( x + 6 ) 1 . The highest power is ( x + 6 ) 1 .
Calculating the LCD To find the LCD, we multiply the LCM of the coefficients by the highest powers of each variable factor: 40 × x 5 × ( x − 6 ) 3 × ( x + 6 ) .
Final Answer Therefore, the least common denominator of the given rational expressions is 40 x 5 ( x − 6 ) 3 ( x + 6 ) . Comparing this with the given options, we see that the correct answer is d.
Examples
When adding or subtracting fractions with polynomial denominators, finding the least common denominator (LCD) is essential. For example, if you are combining terms in a chemical reaction rate equation or designing an electrical circuit where impedances are represented as rational functions, you need to find the LCD to simplify the expressions and solve for unknown variables. Understanding how to find the LCD ensures accurate calculations and efficient problem-solving in these real-world applications.
