In a certain code if 64 is written as 343 and 216 is written as 729, then how is 512 written in that code?

Correct Option/Answer:
The correct answer is 1331.

Step-by-step Explanation:

Step 1: Analyze the given relationships
Let us look at the numbers given in the code and recognize their mathematical patterns:
1. The first pair gives: 64 is written as 343.
Notice that $4^3=64$ and $7^3=343$.
Here, the base changes from 4 to 7 ($4+3=7$), while the power remains 3.

2. The second pair gives: 216 is written as 729.
Notice that $6^3=216$ and $9^3=729$.
Here, the base changes from 6 to 9 ($6+3=9$), while the power remains 3.

Step 2: Identify the pattern/rule
If a number is represented as the cube of an integer $n^3$, then in the coded language, it is transformed into ${(n+3)}^3$.

Step 3: Apply the rule to find the code for 512
First, express 512 as a cube of an integer:
$512=8^3$

Now, add 3 to the base integer (8):
$8+3=11$

Finally, calculate the cube of the new base (11):
${11}^3=11\times 11\times 11=1331$

Therefore, 512 is written as 1331 in that code.