Home > Precalculus > Multiplication of Complex Numbers

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

Answer 1

Sophie Adams

#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#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Lincoln Anderson

To simplify (2\sqrt{-27} \cdot \sqrt{-3}), we first simplify each square root term:

[ \sqrt{-27} = \sqrt{-1 \cdot 3 \cdot 3 \cdot 3} = 3i\sqrt{3} ]

[ \sqrt{-3} = \sqrt{-1 \cdot 3} = i\sqrt{3} ]

Then, we multiply the simplified square roots together:

[ 2\sqrt{-27} \cdot \sqrt{-3} = 2(3i\sqrt{3}) \cdot (i\sqrt{3}) ]

[ = 2(3i \cdot i)(\sqrt{3} \cdot \sqrt{3}) ]

[ = 2(-3)(3) ]

[ = -18 ]

So, (2\sqrt{-27} \cdot \sqrt{-3}) simplifies to (-18).

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.