Find the cube roots of the following using prime factorisation method

Options

A. 5832
B. 19683
C. 9261
D. 405224

Answer

To find the cube roots of the given numbers using the prime factorization method, we will follow these steps:

### Step 1: Prime Factorization
We will start by factorizing each number into its prime factors.

1. **For the number 5, 8, 3, 2:**
- 5 is a prime number.
- 8 = 2 × 2 × 2 = \(2^3\)
- 3 is a prime number.
- 2 is a prime number.
- So, the prime factorization of 5, 8, 3, 2 is:
\[ 5^1 \times 2^3 \times 3^1 \]

2. **For the number 19683:**
- 19683 can be factorized by dividing by 3:
- 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
- So, the prime factorization of 19683 is:
\[ 3^9 \]

3. **For the number 9261:**
- 9261 can be factorized by dividing by 3:
- 9261 ÷ 3 = 3087
- 3087 ÷ 3 = 1029
- 1029 ÷ 3 = 343
- 343 = \(7^3\)
- So, the prime factorization of 9261 is:
\[ 3^3 \times 7^3 \]

4. **For the number 405224:**
- 405224 can be factorized by dividing by 2:
- 405224 ÷ 2 = 202612
- 202612 ÷ 2 = 101306
- 101306 ÷ 2 = 50653 (not divisible by 2 anymore)
- Now, factor 50653 by dividing by 37:
- 50653 ÷ 37 = 1371
- 1371 ÷ 37 = 37
- So, the prime factorization of 405224 is:
\[ 2^3 \times 37^3 \]

### Step 2: Grouping the Factors
Now we will group the prime factors in sets of three to find the cube roots.

1. **For 5, 8, 3, 2:**
- The factors are \(5^1\), \(2^3\), and \(3^1\).
- Grouping gives us:
\[ \sqrt[3]{5^1 \times 2^3 \times 3^1} = 2 \times \sqrt[3]{5 \times 3} = 2 \times \sqrt[3]{15} \]

2. **For 19683:**
- The factors are \(3^9\).
- Grouping gives us:
\[ \sqrt[3]{3^9} = 3^{9/3} = 3^3 = 27 \]

3. **For 9261:**
- The factors are \(3^3\) and \(7^3\).
- Grouping gives us:
\[ \sqrt[3]{3^3 \times 7^3} = 3^{3/3} \times 7^{3/3} = 3 \times 7 = 21 \]

4. **For 405224:**
- The factors are \(2^3\) and \(37^3\).
- Grouping gives us:
\[ \sqrt[3]{2^3 \times 37^3} = 2^{3/3} \times 37^{3/3} = 2 \times 37 = 74 \]

### Final Answers
- The cube root of 5, 8, 3, 2 is \(2 \sqrt[3]{15}\).
- The cube root of 19683 is \(27\).
- The cube root of 9261 is \(21\).
- The cube root of 405224 is \(74\).