Find the least common denominator for these fractions. Enter your answer in the space provided.
$\frac{1}{8}$ and $\frac{1}{7}$
Answer (2)
List the multiples of each denominator: 8 and 7.
Identify the smallest common multiple, which is 56.
Alternatively, find the prime factorizations: 8 = 2 3 and 7 = 7 .
Multiply the highest powers of all prime factors: 2 3 × 7 = 56 .
Explanation
Understanding the Problem We are asked to find the least common denominator (LCD) of the fractions 8 1 and 7 1 . The least common denominator is the smallest multiple that both denominators share.
Listing Multiples To find the LCD, we can list the multiples of each denominator until we find a common multiple. The multiples of 8 are 8, 16, 24, 32, 40, 48, 56, ... The multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56, ...
Prime Factorization Method Alternatively, we can find the prime factorization of each denominator. The prime factorization of 8 is 2 3 . The prime factorization of 7 is 7. The LCD is the product of the highest powers of all prime factors that appear in either factorization. In this case, the prime factors are 2 and 7. The highest power of 2 is 2 3 = 8 , and the highest power of 7 is 7 1 = 7 . Therefore, the LCD is 2 3 × 7 = 8 × 7 = 56 .
Final Answer Thus, the least common denominator of 8 1 and 7 1 is 56 .
Examples
Imagine you are baking a cake and need to measure ingredients using measuring cups of sizes 8 1 cup and 7 1 cup. To easily combine these measurements, you need a common unit. The least common denominator (LCD) helps you find that common unit, which in this case is 56. This means you can think of 8 1 cup as 56 7 cup and 7 1 cup as 56 8 cup, making it easier to add or compare the amounts.
The least common denominator (LCD) of the fractions 8 1 and 7 1 is 56 . This is found either by listing multiples or using prime factorization. In both methods, we confirm that the smallest common multiple of the two denominators is 56.
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