Find the cube roots of the following numbers:
(i)4096 (ii)857375

Answer

To find the cube roots of the numbers 4096 and 857375, we will follow the steps of prime factorization and apply the properties of exponents.

### Step-by-Step Solution

**(i) Finding the cube root of 4096:**

1. **Prime Factorization of 4096:**
- Start dividing 4096 by the smallest prime number, which is 2.
- 4096 ÷ 2 = 2048
- 2048 ÷ 2 = 1024
- 1024 ÷ 2 = 512
- 512 ÷ 2 = 256
- 256 ÷ 2 = 128
- 128 ÷ 2 = 64
- 64 ÷ 2 = 32
- 32 ÷ 2 = 16
- 16 ÷ 2 = 8
- 8 ÷ 2 = 4
- 4 ÷ 2 = 2
- 2 ÷ 2 = 1

So, the prime factorization of 4096 is:
\[ 4096 = 2^{12} \]

2. **Expressing in terms of cubes:**
- We can express \( 2^{12} \) as \( (2^4)^3 \) because \( 2^{12} = (2^4)^3 \).
- Therefore, we can write:
\[ 4096 = (2^4)^3 \]

3. **Finding the cube root:**
- The cube root of 4096 is:
\[ \sqrt[3]{4096} = 2^4 = 16 \]

**Conclusion for (i):**
The cube root of 4096 is **16**.

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**(ii) Finding the cube root of 857375:**

1. **Prime Factorization of 857375:**
- Start dividing 857375 by the smallest prime number, which is 5.
- 857375 ÷ 5 = 171475
- 171475 ÷ 5 = 34295
- 34295 ÷ 5 = 6859
- Now, we need to factor 6859. We can try dividing by 19:
- 6859 ÷ 19 = 361
- 361 ÷ 19 = 19
- 19 ÷ 19 = 1

So, the prime factorization of 857375 is:
\[ 857375 = 5^3 \times 19^3 \]

2. **Expressing in terms of cubes:**
- We can express \( 857375 \) as \( (5 \times 19)^3 \) because \( 857375 = (5 \times 19)^3 \).
- Therefore, we can write:
\[ 857375 = (5 \times 19)^3 \]

3. **Finding the cube root:**
- The cube root of 857375 is:
\[ \sqrt[3]{857375} = 5 \times 19 = 95 \]

**Conclusion for (ii):**
The cube root of 857375 is **95**.

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### Summary of Results:
- The cube root of 4096 is **16**.
- The cube root of 857375 is **95**.