We have seen how the process of simplifying radicals works with numerical values. Now, let's see what happens when algebraic variables are involved.

bullet Let's examine Algebraic Square Roots:
[On this page, the radicand will be non-negative. No negatives under the radical. Also, all variables will be positive.]

statement
A square root is in simplest form when
1. the radicand contains no perfect square factors
2. the radicand is not a fraction
3. there are no radicals in the denominator of a fraction.

Before we begin, take a minute to look at the first table at the right called "Perfect Squares". Notice how variables are perfect squares when their exponents are even numbers. Also, remember the exponent rule, xaxb= xa + b.


expin1
a1

1.
First, we will separate the number value from the algebraic variable. This will give us a chance to examine each for perfect square factors.
                                  rada pic
                                         

2.
Give each factor its own radical sign. a3

3. Reduce the "perfect square" radicals. a22

 

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expin2 anatg4

Separate and find the largest perfect square factors. 
radmath5pic

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expin3 a6

Separate and find the largest perfect square factors. Remember that even numbered exponents are perfect squares.
   radmath61pic

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expin4 radmath7re

        radmathsimp4N
The quotient rule was applied and the perfect square factors found.

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expin5 radexalg4
  
     radmath5re
The denominator is being rationalized by multiplying by the denominator radical value.

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bullet Let's try Algebraic Cube Roots:
Before we begin, take a minute to look at the third table up from the bottom at the right showing algebraic "Perfect Cubes". Notice how variables are perfect cubes when their
exponents are multiples of three.
Again, remember the exponent rule, xaxb= xa + b.

expin6 a8
Remember you are dealing with perfect cubes (not perfect squares). Separate and find the perfect cube factors.

       radamath8pic

bewaresign
When working with cube roots, it is easy to forget to write the index value of 3 on the symbol. Be careful! Without the 3 written in the cube root symbol, your answer will be incorrect, as it represents a different value.

         radcuberoot

Perfect Squares
x2 = x•x
x4 = x2•x2
x6 = x3•x3
x8 = x4•x4
Powers are even.

Product Rule

radthmwhere a ≥ 0, b≥ 0

"The square root of a product is equal to the product of the square roots of each factor."

This theorem allows us to use our method of simplifying radicals.


Quotient Rule

radquotrule
where a ≥ 0, b > 0

"The square root of a quotient is equal to the quotient of the square roots of the numerator and denominator."


Perfect Squares
4 = 2 x 2
9 = 3 x 3
16 = 4 x 4
25 = 5 x 5
36 = 6 x 6
49 = 7 x 7
64 = 8 x 8
81 = 9 x 9
100 = 10 x 10
121 = 11 x 11
144 = 12 x 12
169 = 13 x 13
196 = 14 x 14
225 = 15 x 15

Square Roots
r1
r2
r3
r4
r5
r6
r7
r8
r9
r10
r12
r13
r14
r15

Perfect Cubes
8 = 2 x 2 x 2
27 = 3 x 3 x 3
64 = 4 x 4 x 4
125 = 5 x 5 x 5

Perfect Cubes
x3 = x•x•x
x6 =x2•x2•x2
x9 =x3•x3•x3
x12 =x4•x4•x4
Powers multiples
of 3.


Cube Roots
21
22
23
24

Cube Root Notation
rad cube pic
Remember when working with cube roots to ALWAYS write the index value of 3.

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