The square of a number is the result of a number multiplied by itself. Example 2² (meaning two squared, or 2 x 2) = 4

The square usually has two answers because not only does 2² = 4, so does (-2)² so -2 os also a square root of 4.

It is usually written √4 = ±2 meaning positive or negative 2.

The square root is the value, that when multiplied by itself is the number. Example √4 = 2 because, as we see from the first definition, 2² = 4.

Actually square and square root are the same process, called exponentiation, just raising a number to a different exponential power. "Squaring" a number means to raise that number to the power of 2. The square root means raising it to the power of 1/2. There is a rule about exponents. (a^m)^n = a^(mn). So, [4^(1/2)]^2 = 4^[(1/2)(2)] = 4^1 = 4. They sort of "undo" each other..That's why some people call them opposite operations, or inverse operations..

People are familiar with "squaring" a number, but not so familiar with finding a square root. There's a method kind of like long division that allows one to find the square root, but with the advent of calculators, the method seems to have been forgotten.

One is evolution while the other is involution.

In exponential terminology,

(A) square is the second degree exponent of the base.

e.g. X^2

This means how many times the base will be multiplied by itself.

This is the process of evolution.

(B) square root is the one-half degree exponent of the base.

e.g. X^1/2

This can be expressed in radical form and when simplified will yield a root which when multiplied by itself will yield the base.

This is the process of involution.