Answer :
To find the cube root of the provided numbers using prime factorization, follow these steps:
### a. 5832
1. Perform prime factorization of 5832:
- 5832 is divisible by 2: [tex]\(5832 \div 2 = 2916\)[/tex]
- 2916 is divisible by 2: [tex]\(2916 \div 2 = 1458\)[/tex]
- 1458 is divisible by 2: [tex]\(1458 \div 2 = 729\)[/tex]
- 729 is divisible by 3: [tex]\(729 \div 3 = 243\)[/tex]
- 243 is divisible by 3: [tex]\(243 \div 3 = 81\)[/tex]
- 81 is divisible by 3: [tex]\(81 \div 3 = 27\)[/tex]
- 27 is divisible by 3: [tex]\(27 \div 3 = 9\)[/tex]
- 9 is divisible by 3: [tex]\(9 \div 3 = 3\)[/tex]
- 3 is divisible by 3: [tex]\(3 \div 3 = 1\)[/tex]
2. Gather the prime factors: [tex]\(5832 = 2^3 \times 3^6\)[/tex].
3. To find the cube root, take the cube root of each prime factor raised to a power that is a multiple of 3:
- Cube root of [tex]\(2^3\)[/tex] is [tex]\(2\)[/tex].
- Cube root of [tex]\(3^6\)[/tex] is [tex]\(3^2 = 9\)[/tex].
4. Multiplying these results gives the cube root: [tex]\(2 \times 9 = 18\)[/tex].
### b. 42875
1. Perform prime factorization of 42875:
- 42875 is divisible by 5: [tex]\(42875 \div 5 = 8575\)[/tex]
- 8575 is divisible by 5: [tex]\(8575 \div 5 = 1715\)[/tex]
- 1715 is divisible by 5: [tex]\(1715 \div 5 = 343\)[/tex]
- 343 is divisible by 7: [tex]\(343 \div 7 = 49\)[/tex]
- 49 is divisible by 7: [tex]\(49 \div 7 = 7\)[/tex]
- 7 is divisible by 7: [tex]\(7 \div 7 = 1\)[/tex]
2. Gather the prime factors: [tex]\(42875 = 5^3 \times 7^3\)[/tex].
3. Take the cube root:
- Cube root of [tex]\(5^3\)[/tex] is [tex]\(5\)[/tex].
- Cube root of [tex]\(7^3\)[/tex] is [tex]\(7\)[/tex].
4. Multiplying these results gives the cube root: [tex]\(5 \times 7 = 35\)[/tex].
### c. 74088
1. Perform prime factorization of 74088:
- 74088 is divisible by 2: [tex]\(74088 \div 2 = 37044\)[/tex]
- 37044 is divisible by 2: [tex]\(37044 \div 2 = 18522\)[/tex]
- 18522 is divisible by 2: [tex]\(18522 \div 2 = 9261\)[/tex]
- 9261 is divisible by 3: [tex]\(9261 \div 3 = 3087\)[/tex]
- 3087 is divisible by 3: [tex]\(3087 \div 3 = 1029\)[/tex]
- 1029 is divisible by 3: [tex]\(1029 \div 3 = 343\)[/tex]
- 343 is divisible by 7: [tex]\(343 \div 7 = 49\)[/tex]
- 49 is divisible by 7: [tex]\(49 \div 7 = 7\)[/tex]
- 7 is divisible by 7: [tex]\(7 \div 7 = 1\)[/tex]
2. Gather the prime factors: [tex]\(74088 = 2^3 \times 3^3 \times 7^3\)[/tex].
3. Take the cube root:
- Cube root of [tex]\(2^3\)[/tex] is [tex]\(2\)[/tex].
- Cube root of [tex]\(3^3\)[/tex] is [tex]\(3\)[/tex].
- Cube root of [tex]\(7^3\)[/tex] is [tex]\(7\)[/tex].
4. Multiplying these results gives the cube root: [tex]\(2 \times 3 \times 7 = 42\)[/tex].
### d. 474552
1. Perform prime factorization of 474552:
- 474552 is divisible by 2: [tex]\(474552 \div 2 = 237276\)[/tex]
- 237276 is divisible by 2: [tex]\(237276 \div 2 = 118638\)[/tex]
- 118638 is divisible by 2: [tex]\(118638 \div 2 = 59319\)[/tex]
- 59319 is divisible by 3: [tex]\(59319 \div 3 = 19773\)[/tex]
- 19773 is divisible by 3: [tex]\(19773 \div 3 = 6591\)[/tex]
- 6591 is divisible by 3: [tex]\(6591 \div 3 = 2197\)[/tex]
- 2197 is divisible by 13: [tex]\(2197 \div 13 = 169\ (13 \times 13)\)[/tex]
2. Gather the prime factors: [tex]\(474552 = 2^3 \times 3^3 \times 13^3\)[/tex].
3. Take the cube root:
- Cube root of [tex]\(2^3\)[/tex] is [tex]\(2\)[/tex].
- Cube root of [tex]\(3^3\)[/tex] is [tex]\(3\)[/tex].
- Cube root of [tex]\(13^3\)[/tex] is [tex]\(13\)[/tex].
4. Multiplying these results gives the cube root: [tex]\(2 \times 3 \times 13 = 78\)[/tex].
Thus, the cube roots are:
a. 18
b. 35
c. 42
d. 78
### a. 5832
1. Perform prime factorization of 5832:
- 5832 is divisible by 2: [tex]\(5832 \div 2 = 2916\)[/tex]
- 2916 is divisible by 2: [tex]\(2916 \div 2 = 1458\)[/tex]
- 1458 is divisible by 2: [tex]\(1458 \div 2 = 729\)[/tex]
- 729 is divisible by 3: [tex]\(729 \div 3 = 243\)[/tex]
- 243 is divisible by 3: [tex]\(243 \div 3 = 81\)[/tex]
- 81 is divisible by 3: [tex]\(81 \div 3 = 27\)[/tex]
- 27 is divisible by 3: [tex]\(27 \div 3 = 9\)[/tex]
- 9 is divisible by 3: [tex]\(9 \div 3 = 3\)[/tex]
- 3 is divisible by 3: [tex]\(3 \div 3 = 1\)[/tex]
2. Gather the prime factors: [tex]\(5832 = 2^3 \times 3^6\)[/tex].
3. To find the cube root, take the cube root of each prime factor raised to a power that is a multiple of 3:
- Cube root of [tex]\(2^3\)[/tex] is [tex]\(2\)[/tex].
- Cube root of [tex]\(3^6\)[/tex] is [tex]\(3^2 = 9\)[/tex].
4. Multiplying these results gives the cube root: [tex]\(2 \times 9 = 18\)[/tex].
### b. 42875
1. Perform prime factorization of 42875:
- 42875 is divisible by 5: [tex]\(42875 \div 5 = 8575\)[/tex]
- 8575 is divisible by 5: [tex]\(8575 \div 5 = 1715\)[/tex]
- 1715 is divisible by 5: [tex]\(1715 \div 5 = 343\)[/tex]
- 343 is divisible by 7: [tex]\(343 \div 7 = 49\)[/tex]
- 49 is divisible by 7: [tex]\(49 \div 7 = 7\)[/tex]
- 7 is divisible by 7: [tex]\(7 \div 7 = 1\)[/tex]
2. Gather the prime factors: [tex]\(42875 = 5^3 \times 7^3\)[/tex].
3. Take the cube root:
- Cube root of [tex]\(5^3\)[/tex] is [tex]\(5\)[/tex].
- Cube root of [tex]\(7^3\)[/tex] is [tex]\(7\)[/tex].
4. Multiplying these results gives the cube root: [tex]\(5 \times 7 = 35\)[/tex].
### c. 74088
1. Perform prime factorization of 74088:
- 74088 is divisible by 2: [tex]\(74088 \div 2 = 37044\)[/tex]
- 37044 is divisible by 2: [tex]\(37044 \div 2 = 18522\)[/tex]
- 18522 is divisible by 2: [tex]\(18522 \div 2 = 9261\)[/tex]
- 9261 is divisible by 3: [tex]\(9261 \div 3 = 3087\)[/tex]
- 3087 is divisible by 3: [tex]\(3087 \div 3 = 1029\)[/tex]
- 1029 is divisible by 3: [tex]\(1029 \div 3 = 343\)[/tex]
- 343 is divisible by 7: [tex]\(343 \div 7 = 49\)[/tex]
- 49 is divisible by 7: [tex]\(49 \div 7 = 7\)[/tex]
- 7 is divisible by 7: [tex]\(7 \div 7 = 1\)[/tex]
2. Gather the prime factors: [tex]\(74088 = 2^3 \times 3^3 \times 7^3\)[/tex].
3. Take the cube root:
- Cube root of [tex]\(2^3\)[/tex] is [tex]\(2\)[/tex].
- Cube root of [tex]\(3^3\)[/tex] is [tex]\(3\)[/tex].
- Cube root of [tex]\(7^3\)[/tex] is [tex]\(7\)[/tex].
4. Multiplying these results gives the cube root: [tex]\(2 \times 3 \times 7 = 42\)[/tex].
### d. 474552
1. Perform prime factorization of 474552:
- 474552 is divisible by 2: [tex]\(474552 \div 2 = 237276\)[/tex]
- 237276 is divisible by 2: [tex]\(237276 \div 2 = 118638\)[/tex]
- 118638 is divisible by 2: [tex]\(118638 \div 2 = 59319\)[/tex]
- 59319 is divisible by 3: [tex]\(59319 \div 3 = 19773\)[/tex]
- 19773 is divisible by 3: [tex]\(19773 \div 3 = 6591\)[/tex]
- 6591 is divisible by 3: [tex]\(6591 \div 3 = 2197\)[/tex]
- 2197 is divisible by 13: [tex]\(2197 \div 13 = 169\ (13 \times 13)\)[/tex]
2. Gather the prime factors: [tex]\(474552 = 2^3 \times 3^3 \times 13^3\)[/tex].
3. Take the cube root:
- Cube root of [tex]\(2^3\)[/tex] is [tex]\(2\)[/tex].
- Cube root of [tex]\(3^3\)[/tex] is [tex]\(3\)[/tex].
- Cube root of [tex]\(13^3\)[/tex] is [tex]\(13\)[/tex].
4. Multiplying these results gives the cube root: [tex]\(2 \times 3 \times 13 = 78\)[/tex].
Thus, the cube roots are:
a. 18
b. 35
c. 42
d. 78
