Find the cube root of the following numbers.
(a) 729 (b) 512

Answer

To find the cube roots of the numbers 729 and 512, we can follow these steps:

### Step-by-Step Solution:

**(a) Finding the cube root of 729:**

1. **Factorization of 729:**
- Start by dividing 729 by the smallest prime number, which is 3.
- \( 729 \div 3 = 243 \)
- \( 243 \div 3 = 81 \)
- \( 81 \div 3 = 27 \)
- \( 27 \div 3 = 9 \)
- \( 9 \div 3 = 3 \)
- \( 3 \div 3 = 1 \)
- So, the prime factorization of 729 is \( 3^6 \) (since we divided by 3 a total of 6 times).

2. **Expressing 729 as a cube:**
- We can express \( 729 \) as \( (3^2)^3 \) or \( 9^3 \).

3. **Finding the cube root:**
- The cube root of \( 729 \) can be calculated as:
\[ \sqrt[3]{729} = \sqrt[3]{9^3} = 9 \]

**(b) Finding the cube root of 512:**

1. **Factorization of 512:**
- Start by dividing 512 by 2 (the smallest prime number).
- \( 512 \div 2 = 256 \)
- \( 256 \div 2 = 128 \)
- \( 128 \div 2 = 64 \)
- \( 64 \div 2 = 32 \)
- \( 32 \div 2 = 16 \)
- \( 16 \div 2 = 8 \)
- \( 8 \div 2 = 4 \)
- \( 4 \div 2 = 2 \)
- \( 2 \div 2 = 1 \)
- So, the prime factorization of 512 is \( 2^9 \) (since we divided by 2 a total of 9 times).

2. **Expressing 512 as a cube:**
- We can express \( 512 \) as \( (2^3)^3 \) or \( 8^3 \).

3. **Finding the cube root:**
- The cube root of \( 512 \) can be calculated as:
\[ \sqrt[3]{512} = \sqrt[3]{8^3} = 8 \]

### Final Answers:
- The cube root of 729 is **9**.
- The cube root of 512 is **8**.