Radical & Its Properties

Simplify Radicals

December 2, 2020 (11:30 NN - 1:00 PM)

Learning Objectives

• Understand the rules in solving radicals

• Solve radicals

Recall

Perfect Squares/Cubes

Perfect Square

is a number which is generated by multiplying two equal integers by each other.

Square Root

A square root of a number is one of its two equal factors.

Cube Root

A cube root of a number is one of its three equal factors.

Nth Root of a Number

Let a and b be real numbers and n ≥ 2 be a positive integer.

If a = bⁿ, then b is an n^th root of a.

If n = 2, the root is a square root.

If n = 3, the root is a cube root.

Principal N^th Root

the non-negative square root of a number

Evaluating Expressions involving Radicals

Multiple Choice

 $\sqrt{36}$  

•  $\pm4$  

•  $\pm3$  

•  $\pm9$  

•  $\pm6$  

Multiple Choice

 $\sqrt{225}$  

•  $\pm5$  

•  $\pm10$  

•  $\pm15$  

•  $\pm20$  

Multiple Choice

 $\sqrt{324}$  

•  $\pm16$  

•  $\pm18$  

•  $\pm20$  

•  $\pm22$  

Properties of Radicals

Let a and b be real numbers, variables, or algebraic expressions such that the indicated roots are real numbers, and let m and n be positive integers.

1. $\sqrt{8}\cdot\sqrt{4}$  

= $\sqrt{8\cdot4}$  

= $\sqrt{32}$  
= $\sqrt{16\cdot2}$  
= $4\sqrt{2}$  

Multiple Choice

 $\sqrt{6}\times\sqrt{5}$  

•  $\sqrt{65}$  

•  $\sqrt{30}$  

•  $2\sqrt{5}$  

•  $2\sqrt{3}$  

2. $\frac{\sqrt{9}}{\sqrt{4}}$  

= $\sqrt{\frac{9}{4}}$  

=  $\frac{3}{2}$  

Multiple Choice

 $\frac{\sqrt{144}}{\sqrt{81}}$  

•  $\frac{12}{6}$  

•  $2$  

•  $\frac{12}{9}$  

•  $\frac{4}{3}$  

3. $^{ }\sqrt{\sqrt{81}}$  

= $\left(^{2\cdot2}\sqrt{81}\right)^{ }$ 
= $^4\sqrt{81}$  

= $3$  

Multiple Choice

 $^3\sqrt{\sqrt{64}}$  

•  $2$  

•  $4$  

•  $6$  

•  $8$  

4. $^3\sqrt{125^2}$  

= $\left(^3\sqrt{125}\right)^2$ 
= $\left(5\right)^2$  

= $25$  

Multiple Choice

 $^{^4\sqrt{256^3}}$  

•  $4$  

•  $16$  

•  $48$  

•  $64$  

5. $^3\sqrt{125^3}$  

= $\left(^3\sqrt{125}\right)^3$ 
= $125$ 

Multiple Choice

 $^{^{100}\sqrt{1234^{100}}}$  

•  $1$  

•  $12$  

•  $123$  

•  $1234$  

6. $^3\sqrt{-125^{ }}$  

= $-^3\sqrt{125}$ 
= $-5$ 

Multiple Choice

 $^{^3\sqrt{-216}}$  

•  $-4$  

•  $-6$  

•  $-8$  

•  $-12$  

Simplifying Radicals

 $\sqrt{20}$  

= $\sqrt{4\cdot5}$  
= $2\sqrt{5}$  

 $\sqrt{27}$  

= $\sqrt{9\cdot3}$  
= $3\sqrt{3}$  

Multiple Choice

 $\sqrt{45}$  

•  $2\sqrt{3}$  

•  $2\sqrt{5}$  

•  $3\sqrt{5}$  

•  $9\sqrt{5}$  

 $\sqrt{25x^3}$  

= $\sqrt{25\cdot x^2\cdot x}$  
= $5\cdot\sqrt{x^2\cdot x}$  
= $5\cdot x\cdot\sqrt{x}$  
= $5x\sqrt{x}$  

Multiple Choice

 $^3\sqrt{16y^6}$  

•  $y^2\sqrt{16}$  

•  $2y^2\sqrt{2}$  

•  $2y^2\sqrt{3}$  

•  $2y\sqrt{2y^3}$  

Combining Radicals

 $2\sqrt{48}-3\sqrt{27}$  

= $2\sqrt{16\cdot3}-3\sqrt{9\cdot3}$  
= $2\cdot4\sqrt{3}-3\sqrt{9\cdot3}$  
= $2\cdot4\sqrt{3}-3\cdot3\sqrt{3}$  
= $8\sqrt{3}-9\sqrt{3}$  
= $-\sqrt{3}$  

Multiple Choice

 $4\sqrt{54}+3\sqrt{24}$  

•  $12\sqrt{6}$  

•  $12\sqrt{4}$  

•  $18\sqrt{4}$  

•  $18\sqrt{6}$  

Open Ended

Questions?

Poll

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