What is the square root of -10 times the root of -40?

2 Answers
Lovecraft
Sep 20, 2015

#sqrt(-10)sqrt(-40) = -20#

Explanation:

#sqrt(-10)sqrt(-40) =#
#(sqrt(-10))(sqrt(-40))=#

You can't simply join the roots together, like #sqrt(x)sqrt(y) = sqrt(xy)#, because that formula only works if #x# and #y# aren't both negative. You have to take the negative out of the root first and then multiply then, using the identity #i^2 = -1# where #i# is the imaginary unit, we continue like:

#(sqrt(-1)sqrt(10))(sqrt(-1)sqrt(40))=#
#(isqrt(10))(isqrt(40))=#
#(i^2sqrt(10)sqrt(40))=#
#-sqrt(40*10)=#
#-sqrt(4*100)=#
#-20#

Aron S.
Sep 20, 2015

#sqrt(-10)sqrt(-40) = -20#

Explanation:

Use these two complex number definitions/rules to simplify the expression: #sqrt(-1) = i#, and #i^2 =sqrt(-1)^2= -1#

#sqrt(-10)sqrt(-40) = #
#sqrt(-1*10)sqrt(-1*4*10) = #
#sqrt(-1)sqrt(10)sqrt(-1)sqrt(4)sqrt(10) = #
#sqrt(-1)^2 2 sqrt(10)^2 = #
#-1*2*10 = -20#

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