Math 101 Intermediate Algebra
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Rational Exponents
Chapter 8, Section 2


For the rest of this chapter, it will be assumed that all variables represent non-negative real numbers. So,

 =  a, a    0.


Radical to Exponential Form
For a    0 and n > 0

 =  a1/n

 =  am/n

 =  an/n  =  a


Exponential to Radical Form
For a    0 and n > 0

am/n  = 


Rules of Exponents
The rules of exponents are the same as were given in Chapter 5, section 1.
The rules of exponents apply to rational numbers in the exponents.


Factoring
When factoring out the greatest common factor, take out the smallest exponential power of any factors that are common to all terms.

Example:   x-2 + x2 = x-2(1 + x4) = 1 + x4
x2

Also, look for expressions that are in quadratic form, that are then easily factored. This was done in Chapter 6, section 2.

Example:  
  x4/3 + 2x2/3 + 1
=   (x2/3)2 + 2(x2/3) + 1
  Letting y = x2/3, then
=   y2 + 2y + 1
=   (y + 1)2
  Substituting x2/3 back in for y, gives
=   (x2/3 + 1)2