smalllogi
Simplify Radicals - Algebraic
Square Roots
MathBitsNotebook.com

Topical Outline | Algebra 2 Outline | MathBits' Teacher Resources

Terms of Use   Contact Person: Donna Roberts

divider
If you need a refresher on simplifying radicals with numerical values,
see the Refresher section, Simplifying Radicals.
In this section, we will concentrate on examining algebraic square roots.
Let's see what happens when algebraic variables are involved.


bullet Algebraic Square Roots:
[On this page, all variables will be considered positive.]

statement
Square Roots
Radicals that are simplified have:
1. no fractions left under the radical.
2. no perfect power factors under the radical.
3.
no exponents under the radical greater than the index value.
4. no radicals appearing in the denominator of a fractional answer.

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 rules, xaxb= xa + b and (xa)b = xab.


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

 

divider

expin2 anatg4

Separate and find the largest perfect square factors. 
radmath5pic

divider

expin3 a6

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

divider

expin4 radmath7re

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

divider
expin5 radexalg4
  
     radmath5re
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
The denominator is being "rationalized" (made into a rational number) by multiplying by the denominator's radical value. Both top and bottom are multiplied by this value. In this manner, you are multiplying by "1" and not changing the value of the square root. A perfect square is created in the denominator when multiplied, thus eliminating the radical in the denominator.

divider

bullet So what happens if the radicand is negative?

expin6 higher5
While it is tempting to say that the answer to this simplification is -8ab2, think again. (-8)(-8) ≠ -64
There is no "real number" answer to this problem. We will see how to express an "imaginary" answer in the section on Complex Numbers, since this answer involves the imaginary i..
ANSWER: no real solution

divider

NOTE: The re-posting of materials (in part or whole) from this site to the Internet is copyright violation
and is not considered "fair use" for educators. Please read the "Terms of Use".