`root(3)(103823)=?`

Options

A. 49
B. 51
C. 45
D. 47

Answer

To solve the problem \( \sqrt[3]{103823} \), we will use the method of prime factorization. Here are the steps to find the cube root of 103823:

### Step 1: Prime Factorization
We need to find the prime factors of 103823. We can start by testing divisibility with small prime numbers.

- **Divisibility by 2**: 103823 is odd, so it is not divisible by 2.
- **Divisibility by 3**: The sum of the digits \(1 + 0 + 3 + 8 + 2 + 3 = 17\) is not divisible by 3, so 103823 is not divisible by 3.
- **Divisibility by 5**: The last digit is not 0 or 5, so it is not divisible by 5.
- **Divisibility by 7**: \(103823 \div 7 \approx 14832.42857\) (not divisible).
- **Divisibility by 11**: \(103823 \div 11 \approx 9447.54545\) (not divisible).
- **Divisibility by 13**: \(103823 \div 13 \approx 7986.384615\) (not divisible).
- **Divisibility by 17**: \(103823 \div 17 \approx 6107.235294\) (not divisible).
- **Divisibility by 19**: \(103823 \div 19 \approx 5464.368421\) (not divisible).
- **Divisibility by 23**: \(103823 \div 23 \approx 4518.826087\) (not divisible).
- **Divisibility by 29**: \(103823 \div 29 \approx 3587.827586\) (not divisible).
- **Divisibility by 37**: \(103823 \div 37 \approx 2805.135135\) (not divisible).
- **Divisibility by 43**: \(103823 \div 43 = 2415.186046\) (not divisible).
- **Divisibility by 47**: \(103823 \div 47 = 2209\) (divisible).

Now we can factor 2209:
- \(2209 \div 47 = 47\)

Thus, we find that:
\[ 103823 = 47^3 \]

### Step 2: Applying the Cube Root
Now that we have the prime factorization, we can find the cube root:
\[ \sqrt[3]{103823} = \sqrt[3]{47^3} \]

### Step 3: Simplifying the Expression
Using the property of exponents, we simplify:
\[ \sqrt[3]{47^3} = 47^{3 \cdot \frac{1}{3}} = 47^1 = 47 \]

### Final Answer
Thus, the cube root of 103823 is:
\[ \sqrt[3]{103823} = 47 \]

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