Questions: Assume that the expressions given are denominators of fractions. Find the least common denominator. 7 a+21, a^2+3 a A. 7 a^2+3 B. 7 a^2+21 C. 7 a(a+3) D. 7 a+3

Assume that the expressions given are denominators of fractions. Find the least common denominator.
7 a+21, a^2+3 a
A. 7 a^2+3
B. 7 a^2+21
C. 7 a(a+3)
D. 7 a+3
Transcript text: Assume that the expressions given are denominators of fractions. Find the least common denominator. \[ 7 a+21, a^{2}+3 a \] A. $7 a^{2}+3$ B. $7 a^{2}+21$ C. $7 a(a+3)$ D. $7 a+3$
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Solution

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Solution Steps

Step 1: Factorization of Polynomials

The given expressions are 7a+21 7a + 21 and a2+3a a^2 + 3a . We factor each polynomial:

  1. For 7a+21 7a + 21 : 7a+21=7(a+3) 7a + 21 = 7(a + 3)

  2. For a2+3a a^2 + 3a : a2+3a=a(a+3) a^2 + 3a = a(a + 3)

Step 2: Identify Common Factors

From the factorization, we have:

  • 7(a+3) 7(a + 3)
  • a(a+3) a(a + 3)

The common factor in both expressions is (a+3) (a + 3) .

Step 3: Determine the Least Common Denominator

To find the least common denominator (LCD), we take the product of the highest powers of all factors present in the factorizations:

LCD=7a(a+3) \text{LCD} = 7 \cdot a \cdot (a + 3)

Final Answer

Thus, the least common denominator of the given fractions is: 7a(a+3) \boxed{7a(a + 3)}

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