Square root of 45:

To find the square root of 45, we need to determine a number which, when multiplied by itself, gives us 45.

Prime factorization of 45:
- 45 = 3 x 3 x 5

Method 1: Prime factorization method:
- We can simplify the square root of 45 as the square root of 3 x 3 x 5.
- Taking the square root of 3 x 3 x 5, we get the square root of 3 x the square root of 3 x the square root of 5.
- The square root of 3 is an irrational number and cannot be simplified further.
- Hence, the square root of 45 can be expressed as 3 times the square root of 5.

Method 2: Long division method:
- We can also use the long division method to find the square root of 45.
- We start by grouping the digits of 45 into pairs from right to left, i.e., 4 and 5.
- We find the largest number whose square is less than or equal to 4, which is 2.
- We write 2 as the first digit of the square root and subtract 4 from 45, which gives us 41.
- We bring down the next pair of digits, i.e., 5, which gives us 415.
- We double the first digit of the current result, which is 2, and find the largest number whose product with 20 (i.e., 2 times 10) is less than or equal to 415. This number is 6.
- We write 6 as the second digit of the square root and subtract 36 from 415, which gives us 379.
- We repeat the process until we get the desired accuracy.

Final answer:
- Using either method, we obtain the square root of 45 as 6.7082039325 (approximate value) or 3 times the square root of 5 (exact value).

The square root of 45 is 6 .708