Rad Rat

sqrt(1000)- sqrt(100) +1 =22

The highest square number till 1000 is 31^{2}=961.So there are 31 square numbers lesser than 1000. And there are 9 square numbers lesser than 100. 31-9=22. Therefore,the value of n is 22

10^2 = 100, 32^2 = 1024 >1000. So, the number of n is : 31-10+1 = 22.

The square root of 1000 = 31.6227766

Rounding it down, the perfect square closest to 1000 would be 31*31 = 961

You could attempt 32*32 = 1024 to confirm this.

The root of any integer squared would produce a rational answer.

i.e: 15*15=225 thus root of 225 = 15 (rational)

i.e: 10*10 = 100 thus root of 100 = 10 (rational)

In this case, an irrational answer occurs when the root of a number doesn't produce an integer

i.e: root of 30 = 5.477 (irrational)

i.e: root of 2 = 1.4142 (irrational)

Starting from 1 to 1000,

there are now values 1 to 31 {1, 2, 3, ..., 31} which can produce rational roots after being squared (refer to 2nd line).

We currently have 31 different integers.

Now to eliminate the answers between 1 to 99.

The square root of 99 = 9.949874371

Rounding it down, the perfect square closest to 99 would be 9*9 = 81

So we have values 1 to 9 {1, 2, 3, ..., 9} to be eliminated from our initial pool.

Here, we've got 9 different integers.

Removing the 9 integers from our bigger pool of 31, 31-9 = 22

Our new set of integers now range from 10 to 31 {10, 11, 12, ..., 31} which consists of 22 different integers.

i.e: 10x10 = 100 & 31x31 = 961 (corresponds with the range for roots between 100 and 1000)

the answer is 22

root(n) (smallest)=10 & root(n) (largest) = 31 Or
100<=n<=100 Taking square roots........
10<=root(n)<=31.622
for rational values 10<n<31