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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