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1.2 Simplifying Square Root Radicals
Definitions:
 Whole Numbers (0, 1, 2, 3, 4, …)
 Integers (…, -3, -2, -1, 0 , 1, 2, 3, ….)
ie: Pos and Neg Whole Numbers
 Rational Numbers
o Fractions with Numerator and Denominators both
2 3
Integers ( ,
)
3 5
o Decimals that end (3.45)
o Decimals that repeat (4.1111…)
o Includes all the Integers
 Irrational Numbers
o Decimals that never repeat, and never end
o Includes all the non-perfect square roots ( 3 )
o Note: 4 = 2 (because 2x2=4), so it is not an
irrational number. 4 is called a perfect square
root or a perfect root for short.
o The number PI is considered an irrational number
(3.1415926 etc)
 Real Numbers
o All the Rational Numbers and Irrational Numbers
put together.
Simplifying Perfect Square Roots
 You must know your chart of Perfect Squares from 1 to
100 very well to have success with simplifying square
roots. The first 10 perfect squares are:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100
Since 12 = 1
22 = 4
32 = 9
42 = 16 52 = 25
62 = 36 72 = 49 82 = 64 92 = 81 102 = 100
 16  4
  16 = -4
  16  4 or +4 and -4
 Note:  16 is impossible

x2 = x

9 3

4 2
x 4 = x2
9y 6 = 3y3
3 25  3(5)  15
Simplifying Non-Perfect Square Roots (“Square Root Radicals”)
 The terms “Square Roots” and “Radicals” mean the same
thing.
 The
symbol is called the “radical sign”
 The number written beneath the radical sign is called the
“radicand”.
48 48 is the radicand
 IMPORTANT: A square root is considered to be
unsimplified if the radicand contains a factor that is a
perfect square. That is, the radicand is divisible by a
4, 9, 16, 25, 36, 49, 64, 81, 100, etc.
50 is not simplified because it is divisible by 25.

50  25 2 (We write the perfect root first)
= 5 2 (This is the simplified form! )
Step 1: Look for the biggest factor of 50 that is a perfect
root (25 in this case)
Step 2: Write the radical as a product of the perfect root
and the remaining factor
Step 3: Simplify the perfect root.
(The square root of 25 is 5)
Step 4: Write the remaining factor (Root 2 in this case)
Examples:
Day Two – Variables in the Radicand
 We can do the same for variables in the radicand.
x5  x4 x
= x2 x
Note: Radicands with EVEN exponents have PERFECT
ROOTS . Radicands with exponents bigger than
1 must be simplified! Radicands with ODD
exponents bigger than 1 will become mixed radicals.
x11
  x10 x
 x5 x
x7
  x6 x
 x3 x
50 x13
  25 x12 2 x
 5x6 2 x

200x 5 y 6 = 100x 4 y 6 2x
 10 x 2 y 3 2 x

1
1
8x 3 =
4x 2 2x
2
2
1
= 2x  2x
2
=
x
2x
(Note ½ times 2 = 1)
Changing from Mixed Radicals to Entire Radicals

200 is called an entire radical.
 10 2 is called a mixed radical.
 To go from a “mixed” to an “entire” radical, we work
backwards from our simplifying steps:
10 2 = 100 2 (Note 10 = 100 )
= 200
(or 10 times 10 times 2 = 200)
Assignment:
Page 14 #1 to 59 ODDS only
Page 15 #61 to #143 ODDS only