1. 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

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

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

4. Simplify or Reduce a Radical by Finding Perfect Squares Reduce 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.

5. 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 Simplify or Reduce a Radical by Finding Prime Factors Reduce

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

7. Simplify Each Expression Simplify Method 1: Finding Perfect Squares Method 2: Factor into Prime Numbers

8. A radical expression is in simplest form when all the following are true.

9. Simplify or Reduce a Radical in Fraction 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. Write in simplest form.

10. Simplify or Reduce a Radical Simplify the expression

11. 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:

12. Natural Numbers 1, 2, 3, … 1 2 3 4 5 6 7 8 9 10 11 Whole Numbers 0, 1, 2, 3, … 0 1 2 3 4 5 6 7 8 9 10 11 Integers... -3, -2, -1, 0, 1, 2, 3, … -3 -2 -1 0 1 2 3 4 Rational 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….) Irrational Numbers Any number that can not be written in the form of . (Nonrepeating & non-terminating) Real Number System

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