How do you multiply the square root of 3 times the square root of 5?

1 Answer
George C.
Oct 13, 2015

If #a, b >= 0# then #sqrt(a)sqrt(b) = sqrt(ab)#, so

#sqrt(3)sqrt(5) = sqrt(3*5) = sqrt(15)#

Explanation:

#(sqrt(a)sqrt(b))^2 = sqrt(a)sqrt(b)sqrt(a)sqrt(b) = sqrt(a)sqrt(a)sqrt(b)sqrt(b) = ab#

So #sqrt(a)sqrt(b)# is always a square root of #ab#, but is it #sqrt(ab)# or #-sqrt(ab)#?

If #a, b >= 0# then #sqrt(a)#, #sqrt(b)# and #sqrt(ab)# are all Real and non-negative, so #sqrt(a)sqrt(b) = sqrt(ab)#

But if #a, b < 0# then #sqrt(a)sqrt(b) != sqrt(ab)# as we can see:

#1 = sqrt(1) = sqrt(-1*-1) != sqrt(-1)*sqrt(-1) = -1#

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