How do you simplify # 2sqrt(-27) • sqrt(-3)#?

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Answer 1 · 2018-03-02T14:02:14

#2sqrt(-27)sqrt(-3) = -18#

It is interesting to note what you should not do.

Note that:

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

if at least one of #a# and #b# is non-negative.

In our example we have #a = -27 < 0# and #b = -3 < 0# and we might erroneously deduce:

#color(red)(cancel(color(black)(2sqrt(-27)sqrt(-3) = 2sqrt((-27)(-3)) = 2sqrt(81) = 2*9 = 18)))#

Instead, note that if #a < 0# then #sqrt(a) = i sqrt(-a)#

So we have:

#2sqrt(-27)sqrt(-3) = 2i sqrt(27) i sqrt(3) = 2i^2 sqrt(27)sqrt(3) = -2sqrt(81) = -2*9 = -18#