Simplifying Radicals and Imaginary Numbers

The Imaginary Numbers

What are imaginary numbers?

• Equations such as

   $x^2+1=0$  have no real solutions, so mathematicians defined imaginary numbers to represent their solutions.

•  $i=\sqrt{-1}$    This is useful when working with square roots of negative numbers

• An imaginary number is written in the form  $bi$  , where  $i$   is the imaginary part and b is the real part

Let's work one together!

 $\sqrt{-5}$  

• Rewrite  $\sqrt{-x}$   as  $\sqrt{-1\times x}$  

• Break x down if it is not a perfect square

• Simplify the radical, recall that  $i=\sqrt{-1}$  

Multiple Choice

Your Turn!

$$\sqrt{-9}$$  

• $$\pm3$$  

• $$\pm3i$$  

• $$\pm9i$$  

• $$\pm9$$  
  

Multiple Choice

$$x^2+81=0$$  

• $$\pm\sqrt{9i}$$  

• $$\pm9$$  

• $$\pm9i$$  

• $$\pm3i$$  

Now let's look at the powers of

 $i$  

•  $i=\sqrt{-1}$  

•  $i^2=-1$  

•  $i^3=-i$  

•  $i^4=1$  

 $i^{15}$  

How do you think you would solve this?

Multiple Choice

Your Turn!

$$i^{22}$$  

• $$\sqrt{-1}$$  

• $$-1$$  

• $$-i$$  

• 1

Product of Imaginary Numbers

 $\sqrt{-18}\cdot\sqrt{-10}$  

First take the  $i$   out of the radical then solve

Multiple Choice

$$\sqrt{-8}\cdot\sqrt{24}$$  

• $$i\sqrt{24}$$  

• $$8\sqrt{3}$$  

• $$8i\sqrt{3}$$  

• $$60i$$  

Multiple Choice

$$4i\cdot7i$$  

• $$28i$$  

• -28

• 28

• $$14i$$  

Multiple Choice

$$\left(2i\right)^3\cdot\left(5i\right)$$  

• 40

• $$40i$$  

• 10

• $$10i$$