Question

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

Answer 1

`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`

Answer 2

`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`