Theorem: for all non-negative real numbers a and b, SQRT(a*b) = SQRT(a)*SQRT(b)
Proof: suppose a and b are any non negative real numbers. Then there exists unique non negative real numbers m and n such that a = m
^{2} and b = n
^{2}. Then
a*b = m
^{2}*n
^{2} by substitution
= (m*n)
^{2} by laws of exponents.
Then by taking the square root of both sides
SQRT(a*b) = m*n
but because a = m
^{2} and b = n
^{2}, it follows that m = SQRT(a) and n = SQRT(b), and so , by substitution
SQRT(a*b) = SQRT(a)*SQRT(b)
Q.E.D
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