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QUESTION

# What is the square root of 64?

The principal square root of ##64## is:

##sqrt(64) = 8##

The other (non-principal) square root is:

##-sqrt(64) = -8##

##64## has two square roots, namely ##8## and ##-8##, since:

##8^2 = (-8)^2 = 64##

When we say "the square root", what is usually intended is "the principal square root", which in the case of the Real square root of a positive number is the positive one.

Any non-zero number ##n## has two square roots. In order to distinguish between them, we call one the "principal" square root, which in the case of ##n > 0## means the positive one.

##color(white)()##Complex footnote

If ##n < 0## then it has two Complex non-Real square roots:

##+-i sqrt(-n)##

In this case we call ##i sqrt(-n)## the principal square root and ##-i sqrt(-n)## the non-principal one.

For example:

##sqrt(-64) = 8i## is the principal square root.

##-sqrt(-64) = -8i## is the other square root.

Note that ##8i## is not "positive". Unlike Real numbers, Complex numbers are not ordered, but for pure imaginary square roots we choose the one with the positive imaginary part and call it "principal".