Find the cube root of the following numbers by expressing each number as the product of primes:
(i) 1728 (ii) 4096 (iii) 27000.

Answer

To find the cube root of the given numbers by expressing each number as the product of primes, we will follow these steps for each number:

### (i) Finding the cube root of 1728

1. **Factor 1728 into prime factors:**
- Since 1728 is even, we start dividing by 2:
- \( 1728 \div 2 = 864 \)
- \( 864 \div 2 = 432 \)
- \( 432 \div 2 = 216 \)
- \( 216 \div 2 = 108 \)
- \( 108 \div 2 = 54 \)
- \( 54 \div 2 = 27 \)
- Now, 27 is not even, so we divide by 3:
- \( 27 \div 3 = 9 \)
- \( 9 \div 3 = 3 \)
- \( 3 \div 3 = 1 \)
- The prime factorization of 1728 is:
\[ 1728 = 2^6 \times 3^3 \]

2. **Calculate the cube root:**
- The cube root of \( 1728 \) can be expressed as:
\[ \sqrt[3]{1728} = \sqrt[3]{2^6 \times 3^3} \]
- Using the property of exponents:
\[ = 2^{6/3} \times 3^{3/3} = 2^2 \times 3^1 = 4 \times 3 = 12 \]

### (ii) Finding the cube root of 4096

1. **Factor 4096 into prime factors:**
- Since 4096 is even, we divide by 2 repeatedly:
- \( 4096 \div 2 = 2048 \)
- \( 2048 \div 2 = 1024 \)
- \( 1024 \div 2 = 512 \)
- \( 512 \div 2 = 256 \)
- \( 256 \div 2 = 128 \)
- \( 128 \div 2 = 64 \)
- \( 64 \div 2 = 32 \)
- \( 32 \div 2 = 16 \)
- \( 16 \div 2 = 8 \)
- \( 8 \div 2 = 4 \)
- \( 4 \div 2 = 2 \)
- \( 2 \div 2 = 1 \)
- The prime factorization of 4096 is:
\[ 4096 = 2^{12} \]

2. **Calculate the cube root:**
- The cube root of \( 4096 \) can be expressed as:
\[ \sqrt[3]{4096} = \sqrt[3]{2^{12}} \]
- Using the property of exponents:
\[ = 2^{12/3} = 2^4 = 16 \]

### (iii) Finding the cube root of 27000

1. **Factor 27000 into prime factors:**
- We can express 27000 as:
\[ 27000 = 27 \times 1000 \]
- Factor 27:
\[ 27 = 3^3 \]
- Factor 1000:
\[ 1000 = 10^3 = (2 \times 5)^3 = 2^3 \times 5^3 \]
- Therefore, the prime factorization of 27000 is:
\[ 27000 = 3^3 \times 2^3 \times 5^3 \]

2. **Calculate the cube root:**
- The cube root of \( 27000 \) can be expressed as:
\[ \sqrt[3]{27000} = \sqrt[3]{3^3 \times 2^3 \times 5^3} \]
- Using the property of exponents:
\[ = 3^{3/3} \times 2^{3/3} \times 5^{3/3} = 3^1 \times 2^1 \times 5^1 = 3 \times 2 \times 5 = 30 \]

### Summary of Results
- Cube root of 1728 is **12**.
- Cube root of 4096 is **16**.
- Cube root of 27000 is **30**.