Simplifying Radicals and Roots

Simplifying Radicals

Solving by taking Square Roots

Multiple Choice

Which of the following is a perfect square?

• $$\sqrt{8}$$

• $$\sqrt{49}$$

• $$\sqrt{18}$$

• $$\sqrt{125}$$

To Simplify a Square Root is the process of evaluating the perfect squares that are factors.

For example:

 $\sqrt{8}=\sqrt{4}\times\sqrt{2}$  

 $\sqrt{8}=2\sqrt{2}$  

Multiple Choice

Simplifying a Square Root:

$$\sqrt{perfect\ square}\times\sqrt{non\ perfect\ square}$$   $$example:\ \ \sqrt{45}$$  

• $$9\sqrt{5}$$  

• $$5\sqrt{9}$$  

• $$3\sqrt{5}$$  

• $$5\sqrt{3}$$  

Multiple Choice

$$Simplify:\ \ \sqrt{20}$$
 

• $$2\sqrt{10}$$  

• $$4\sqrt{5}$$  

• $$10\sqrt{2}$$  

• $$2\sqrt{5}$$  

Solving Quadratics by taking Square Roots:

Solve (there will be two solutions):
 $\left(x+3\right)^2=100$  

 $\left(x+6\right)^2=45$  

Answer should be in simplified radical form.

Multiple Select

$$Solve:\ \left(x-8\right)^2=28$$  
Next steps:

• Square both sides

• Take the square root of both sides

Multiple Select

$$\sqrt{\left(x-8\right)^2}=\sqrt{28}$$  

• $$x^2-64=14$$  

• $$x-8=\pm\sqrt{28}$$  

Multiple Select

$$x-8=\pm\sqrt{28}$$  

• $$x=8\pm4\sqrt{7}$$  

• $$x=8\pm7\sqrt{2}$$  

• $$x=8\pm2\sqrt{7}$$  

• $$x=-8\pm2\sqrt{7}$$