Imaginary numbers are numbers that cannot exist. They can be identified by the letter i. They are just like normal numbers, only they have an i connected with them. Imaginary numbers are written something like 5i or 3 + 2i.

 

Remember that square roots were trying to find out what number times itself would produce the number underneath the radical. By the rules of multiplication, if two positive numbers or two negative numbers are multiplied, the result will always be a positive number. Therefore if one multiplies a number by itself it will produce a positive answer. This is where imaginary numbers come into play. By the definition of the letter i, squaring it (multiply it by itself), will result in –1.

 

 

Now under the same definition,

 

= .

 

In order to solve problems involving negative radicals, treat them just like normal square root problems. In other words, break it down into easy square roots (perfect squares, -1, and variables with even numbered exponents).

 

Example

 

- 4 can also be written as – 1(4).

 

=and = .

 

Answer!

 

Note- Always write the number (coefficient) before any letter.

 

 

 

 

Example

-98 can be written as –1(49)2

and =

 

 

 

Substitute in =7

 

And = . Combine what is left under the radicals into a single radical (2 and x).

 

Answer!

 

 

 

 

 

Example

 

= .

 

Answer!

 

 

 

 

Example

 

=

 

=

 

Multiply.

 

Answer!

 

 

 

 

Example

 

 

=

 

By the definition of exponents.

 

Since =

 

= , by definition of i

 

.

 

Simplify.

 

Answer!

 

 

 

Example

 

-2 is –1(2) and –6 is -1(6)

 

=

 

 

= and =

 

Also =

 

=

 

Multiplying anything by 1 does not change anything so just drop the 1.

 

=

 

 

=

 

 

Answer!