 Simplify Radicals - Algebraic Square Roots MathBitsNotebook.com Terms of Use   Contact Person: Donna Roberts 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. Algebraic Square Roots: 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.  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. 2.
Give each factor its own radical sign. 3. Reduce the "perfect square" radicals.    Separate and find the largest perfect square factors.    Separate and find the largest perfect square factors. Remember that even numbered exponents are perfect squares.     The quotient rule was applied and the perfect square factors found.    Perfect Squares x2 = x•x x4 = x2•x2 x6 = x3•x3 x8 = x4•x4 Powers are even.

 Product Rule where 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 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              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.  So what happens if the radicand is negative?  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.. 