What will come in the place of question mark (?) in following questions?
`root3(46656)+ root3(9261)`= ?

Options

A. 57
B. 56
C. 58
D. 66

Answer

To solve the expression \( \sqrt[3]{46656} + \sqrt[3]{9261} \), we will find the cube roots of both numbers step by step.

### Step 1: Calculate \( \sqrt[3]{46656} \)

To find the cube root of 46656, we can factor it:

1. Start dividing by the smallest prime number, which is 2:
- \( 46656 \div 2 = 23328 \)
- \( 23328 \div 2 = 11664 \)
- \( 11664 \div 2 = 5832 \)
- \( 5832 \div 2 = 2916 \)
- \( 2916 \div 2 = 1458 \)
- \( 1458 \div 2 = 729 \) (729 is no longer divisible by 2)

2. Now divide by 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 46656 is:
\[ 46656 = 2^6 \times 3^6 \]

Now, to find the cube root:
\[ \sqrt[3]{46656} = \sqrt[3]{(2^6 \times 3^6)} = 2^{6/3} \times 3^{6/3} = 2^2 \times 3^2 = 4 \times 9 = 36 \]

### Step 2: Calculate \( \sqrt[3]{9261} \)

Next, we find the cube root of 9261:

1. Start dividing by 3:
- \( 9261 \div 3 = 3087 \)
- \( 3087 \div 3 = 1029 \)
- \( 1029 \div 3 = 343 \) (343 is no longer divisible by 3)

2. Now divide by 7:
- \( 343 \div 7 = 49 \)
- \( 49 \div 7 = 7 \)
- \( 7 \div 7 = 1 \)

So, the prime factorization of 9261 is:
\[ 9261 = 3^3 \times 7^3 \]

Now, to find the cube root:
\[ \sqrt[3]{9261} = \sqrt[3]{(3^3 \times 7^3)} = 3^{3/3} \times 7^{3/3} = 3 \times 7 = 21 \]

### Step 3: Add the two cube roots

Now we can add the results from Steps 1 and 2:
\[ \sqrt[3]{46656} + \sqrt[3]{9261} = 36 + 21 = 57 \]

### Final Answer

Thus, the value that comes in place of the question mark (?) is:
\[ \boxed{57} \]

---