Find the cube root of the following:
(a) 4913 (b) 19683

Answer

To find the cube roots of the numbers 4913 and 19683, we will break down the process step by step.

### Step-by-Step Solution:

**(a) Finding the cube root of 4913:**

1. **Prime Factorization of 4913:**
- Start dividing 4913 by the smallest prime number, which is 2. Since 4913 is odd, we move to the next prime number, which is 3.
- 4913 is not divisible by 3. Next, we try 5, which also does not divide evenly.
- Next, we try 7, 11, and finally 17.
- Dividing 4913 by 17 gives us 289 (4913 ÷ 17 = 289).
- Now, we factor 289. We find that 289 is 17 × 17 (or 17²).
- Therefore, we can express 4913 as:
\[ 4913 = 17 × 17 × 17 = 17^3 \]

2. **Finding the Cube Root:**
- The cube root of a number is the value that, when multiplied by itself three times, gives the original number.
- Since \( 4913 = 17^3 \), the cube root can be calculated as:
\[ \sqrt[3]{4913} = \sqrt[3]{17^3} = 17 \]

**Conclusion for (a):**
The cube root of 4913 is **17**.

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**(b) Finding the cube root of 19683:**

1. **Prime Factorization of 19683:**
- Start dividing 19683 by 3 (the smallest prime number).
- 19683 ÷ 3 = 6561.
- 6561 ÷ 3 = 2187.
- 2187 ÷ 3 = 729.
- 729 ÷ 3 = 243.
- 243 ÷ 3 = 81.
- 81 ÷ 3 = 27.
- 27 ÷ 3 = 9.
- 9 ÷ 3 = 3.
- 3 ÷ 3 = 1.
- We divided by 3 a total of 9 times, so we can express 19683 as:
\[ 19683 = 3^9 \]

2. **Finding the Cube Root:**
- To find the cube root, we can use the property of exponents:
\[ \sqrt[3]{3^9} = 3^{9/3} = 3^3 = 27 \]

**Conclusion for (b):**
The cube root of 19683 is **27**.

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### Final Answers:
- (a) The cube root of 4913 is **17**.
- (b) The cube root of 19683 is **27**.