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