Introduction:
To find the smallest number by which 46305 can be multiplied to make it a perfect cube, we need to follow a step-by-step approach.

Steps:
1. Factorize the given number:
We need to factorize the given number 46305 into its prime factors.
46305 = 3 x 5 x 3087
Now, we need to check the powers of each prime factor.
3 is already a perfect cube.
5 needs to be multiplied by 5 to make it a perfect cube.
3087 needs to be factorized further.
3087 = 3 x 1029
1029 = 3 x 343
343 = 7 x 7 x 7
Therefore, the prime factorization of 46305 is:
46305 = 3^4 x 5 x 7^3

2. Find the smallest number to make each prime factor a perfect cube:
To make each prime factor a perfect cube, we need to find the smallest number to multiply each prime factor.
For 3, it is already a perfect cube, so we don't need to multiply it.
For 5, we need to multiply it by 5 to make it a perfect cube.
For 7, we need to multiply it by 7 to make it a perfect cube.

3. Multiply all the numbers to get the final result:
Now, we need to multiply all the numbers that we got in step 2.
Smallest number to make 46305 a perfect cube = 3 x 5 x 7 = 105

Conclusion:
Therefore, the smallest number by which 46305 can be multiplied to make it a perfect cube is 105.

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