4. Real Number System and Radicals SPI 11A: order a set of rational and irrational numbers SPI 12B: simplify a radical
Objectives:
Investigate the Real Number System
Review operations on radical expressions

5. Relate Perfect Squares to Radicals Area of a Square (A = l · w)2 2 3 3 4 4

6. Methods for Simplifying Radical Expressions
Two Methods for Simplifying Radical Expressions:
1. Simplify by finding perfect squares
2. Simplify by creating a factor tree

7. Simplify or Reduce a Radical by Finding Perfect Squares
STEP 1. Write the number appearing under your radical as the product (multiplication) of the perfect square.
STEP 2. Write the values under the radical separately.
STEP 3. Simplify the expression.

8. Simplify or Reduce a Radical by Finding Prime Factors
Step 1. 
Factor the number under the radical sign into prime numbers
48 = 2 ∙ 24
2 ∙ 12
2 ∙ 6
2 ∙ 3
Prime numbers of 48 = 2 ∙ 2 ∙ 2 ∙ 2 ∙ 3

9. Simplify or Reduce a Radical by Finding Prime Factors
Step 2. 
Group prime numbers into squares
Step 3. 
Reduce the radical from the perfect squares
Step 4. 
Simplify the radical.

10. Simplify Each Expression
Try Now!
Simplify Each Expression
Simplify
Method 1: Finding Perfect Squares
Method 2: Factor into Prime Numbers

11. The number under the radical sign has
A radical expression is in simplest form when all the following are true.
The number under the radical sign has
no perfect square factors other than 1.
The number under the radical sign does
not contain a fraction.
A denominator does not contain a radical
expression.

12. Simplify or Reduce a Radical in Fraction Form
Write in simplest form.
Step 1. 
Rewrite the single radical as a quotient.
Step 2. 
Simplify, if possible.
Step 3. 
Multiply by a form of 1 to rationalize the denominator.
Do not leave a radical in the denominator.
Step 4. 
Simplify.

13. Simplify or Reduce a Radical
Simplify the expression

14. Enter the Real-world of using radicals
Using the Pythagorean Theorem, find the length of the skateboard ramp.
1 foot
7 foot
Use the Pythagorean Theorem:

15. (Can be terminating, such as 6.27 .. Or .. Repeating like 8.222….)
Real Number System
Rational Numbers
Irrational Numbers
Any number that can be written in the form  where a and b are integers and b is not equal to 0.
(Can be terminating, such as 6.27
.. Or .. Repeating like 8.222….)
Any number that can not be written in the form of .
(Nonrepeating & non-terminating)
Natural Numbers
1, 2, 3, …
Whole Numbers
0, 1, 2, 3, …
Integers
, -2, -1, 0, 1, 2, 3, …

17. A number that CANNOT be written as a ratio.
Irrational Numbers and the Number Line
A number that CANNOT be written as a ratio.
Estimate the location of the following on a number line: