real number system and radicals spi 11a: order a set of rational and irrational numbers spi 12b:...

Real Number System and Radicals SPI 11A: order a set of rational and irrational numbersSPI 12B: simplify a radical

Objectives:• Investigate the Real Number System• Review operations on radical expressions

Relate Perfect Squares to RadicalsArea of a Square (A = l · w)

2

2

3

3

4

4

Methods for Simplifying Radical Expressions

Two Methods for Simplifying Radical Expressions:

1. Simplify by finding perfect squares

2. Simplify by creating a factor tree

Simplify or Reduce a Radical by Finding Perfect Squares

Reduce 48

STEP 1. Write the number appearing under your radical as the product (multiplication) of the perfect square.

31648

STEP 2. Write the values under the radical separately.

31648

STEP 3. Simplify the expression.

3448

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

48Reduce

Step 2.  Group prime numbers into squares

Step 3.  Reduce the radical from the perfect squares

Step 4. Simplify the radical.

48 2 2 2 2 3

2 22 2 3

32 2

4 3

Simplify or Reduce a Radical by Finding Prime Factors

Simplify Each Expression

Simplify 72Method 1: Finding Perfect Squares Method 2: Factor into Prime Numbers

72 2 36

2 36

6 2

72 2 36

2 18

2 9

3 3

72 2 2 2 3 3 2 272 2 2 3

72 2 3 2

= 6 2

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

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.

4

3Write in simplest form.

4 4

3 3

4 2 4 2 2 2

3 3 3 3 3or

2 3

3 3

2 3 2 3

33 3

Simplify or Reduce a Radical

Simplify the expression

5 10

5 10 5 10

5 2 5

5 2

Enter the Real-world of using radicalsUsing the Pythagorean Theorem, find the length of the skateboard ramp.

1 foot

7 foot

Use the Pythagorean Theorem:2 2 2c a b

2 2 21 7c 2 50c

50c2 25c

5 2c

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 NumbersAny 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

Irrational Numbers and the Number LineA number that CANNOT be written as a ratio.

Estimate the location of the following on a number line:

2 3