Fractional exponents are manipulated exactly like integral exponents, so if you know those rules you are in good shape.

There are two sets of rules:

• Rules  for Combining (Adding and Subtracting) terms that have exponents
  

  • The only terms that can be combined are LIKE terms.  The exponent of a base determines what kind of term you have.  Therefore, in order to combine terms that have exponents, the exponents must be the same,
    

  • Since the exponent determines what kind of term you have, you cannot change the exponent without changing the kind of term it is.
    

  • Bottom Line:  When combining terms that have integral exponents, you never change the exponents.  Rather, you simply combine all the terms that have the same base and the same exponent.  How many of them there are is determined by their multipliers ("coefficients").
    

  

  Here the 3x³ and the 2x³  are like terms, and the 7x⁴  and the -3x⁴ are like terms.  The 3x³ and the 2x³ combine to make 5x³.   The 7x⁴  and the -3x⁴  combine to make 4x⁴.  Therefore, the combining of the example expression is:

                                

  

• Rules for Multiplying/Dividing terms that have exponents
  

  • The product of like bases is the base raised to the sum of the exponents.                    
    
    
    
    

  • The quotient of like bases is the base raised to the power of the numerator minus the power of the denominator.
    
    
    

  • A power raised again to another power is the base raised to the product of the two powers.
    
    
    

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