Example:

Combine the subsequent imaginary numbers:

Solution:

√-16 + √-36 -√-49 - √-1 =

√-16 + √-36 -√-49 - √-1 = 4i +6i -7i-i

= 10i -8i

= 2i

Therefore the result is 2i = 2√-1 = √-4

Imaginary numbers are multiplied/divided by writing them using the imaginary unit i, and after that multiplying or dividing them such as algebraic terms.  Therefore, there are various primary relationships that must also be used to multiply/divide imaginary numbers.

i² = (i)(i) = (√-1)(√-1) = -1

i³  = (i²)(i) = (-1)(i) = -i

i⁴  = (i²)(i²) = (-1)(-1) = +1

Using these basic relationships, for example, (√-25)(√-4) equals (5i)(2i) which equals 10i². But, i² equals -1.  Thus, 10i² equals (10)(-1) which equals -10.

Any square root has two roots, i.e., a statement x²  = 25 is a quadratic and has roots

x = ±5 since +52  = 25 and (-5) x (-5) = 25.

Similarly,

√-25 = 5i

√-4 =±2i

And

√-25 √-4 =±10.