All prime numbers have irrational number square roots, so if you try to find the square root of a non-perfect square number use them to simplify it. For example,
±√125 = ±√25*5 = ±5√5 (when you want to show both the square roots)
√72 = √36*2 = 6√2
√-27 = √-9*3 = 3i√3
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Imaginary numbers are only ever used when you are using the square roots of negative numbers. The square root of -1 is i. You may find imaginary numbers when you are finding roots of equations.
Fne if they are sufficiently far apart. Otherwise, you may be better off squaring all the numbers. The smaller numbers will still have the smaller squares and at least you won't have irrational numbers to deal with.
Any number that can't be expessed as a fraction is an irrational number as for example the square root of 4.5
There may be many easier and better ways, but here's how I would do it: -- Square the first given irrational number. -- Square the second irrational number. -- Pick a nice ugly complicated decimal between the two squares. -- Take the square root of the number you picked. It's definitely between the two given numbers, and it would be a miracle if it's not irrational.
72 = 49 and 82 = 64. So, the square root of any integer between these two numbers, for example, sqrt(56), is irrational.