Sure, let's find the least common denominator (LCD) for each group of fractions step-by-step.
### Part a: [tex]\( \frac{1}{6} \)[/tex] and [tex]\( \frac{7}{8} \)[/tex]
To find the least common denominator for [tex]\( \frac{1}{6} \)[/tex] and [tex]\( \frac{7}{8} \)[/tex], we need to find the least common multiple (LCM) of the denominators 6 and 8.
The prime factorizations are:
- 6 = 2 × 3
- 8 = 2³
The LCM is found by taking the highest power of each prime that appears in the factorizations:
- Highest power of 2 is 2³ = 8
- Highest power of 3 is 3¹ = 3
Thus, the LCM of 6 and 8 is 8 × 3 = 24.
So, the least common denominator for [tex]\( \frac{1}{6} \)[/tex] and [tex]\( \frac{7}{8} \)[/tex] is 24.
### Part b: [tex]\( \frac{3}{4} \)[/tex] and [tex]\( \frac{7}{10} \)[/tex]
To find the least common denominator for [tex]\( \frac{3}{4} \)[/tex] and [tex]\( \frac{7}{10} \)[/tex], we need to find the LCM of the denominators 4 and 10.
The prime factorizations are:
- 4 = 2²
- 10 = 2 × 5
The LCM is:
- Highest power of 2 is 2² = 4
- Highest power of 5 is 5¹ = 5
Thus, the LCM of 4 and 10 is 4 × 5 = 20.
So, the least common denominator for [tex]\( \frac{3}{4} \)[/tex] and [tex]\( \frac{7}{10} \)[/tex] is 20.
### Part c: [tex]\( \frac{7}{12}, \frac{3}{8}, \)[/tex] and [tex]\( \frac{11}{36} \)[/tex]
To find the least common denominator for [tex]\( \frac{7}{12}, \frac{3}{8}, \)[/tex] and [tex]\( \frac{11}{36} \)[/tex], we need to find the LCM of the denominators 12, 8, and 36.
The prime factorizations are:
- 12 = 2² × 3
- 8 = 2³
- 36 = 2² × 3²
The LCM is:
- Highest power of 2 is 2³ = 8
- Highest power of 3 is 3² = 9
Thus, the LCM of 12, 8, and 36 is 8 × 9 = 72.
So, the least common denominator for [tex]\( \frac{7}{12}, \frac{3}{8}, \)[/tex] and [tex]\( \frac{11}{36} \)[/tex] is 72.
### Part d: [tex]\( \frac{8}{15}, \frac{11}{30}, \)[/tex] and [tex]\( \frac{3}{5} \)[/tex]
To find the least common denominator for [tex]\( \frac{8}{15}, \frac{11}{30}, \)[/tex] and [tex]\( \frac{3}{5} \)[/tex], we need to find the LCM of the denominators 15, 30, and 5.
The prime factorizations are:
- 15 = 3 × 5
- 30 = 2 × 3 × 5
- 5 = 5
The LCM is:
- Highest power of 2 is 2¹ = 2
- Highest power of 3 is 3¹ = 3
- Highest power of 5 is 5¹ = 5
Thus, the LCM of 15, 30, and 5 is 2 × 3 × 5 = 30.
So, the least common denominator for [tex]\( \frac{8}{15}, \frac{11}{30}, \)[/tex] and [tex]\( \frac{3}{5} \)[/tex] is 30.
