GMAT Maths Algebra - Exponents

  • Exponents are a "short cut" method of showing a number is multiplied by itself. 
  • Exponents can be shown to different ways.  Example: 
    •         x2 or x^2
  • Know the difference between -x^y and (-x)^y.  Example: 
    •         -34 = -(3)(3)(3)(3) = -81
            (-3)4 = (-3)(-3)(-3)(-3) = 81

    The evaluation of expressions containing exponents is very straightforward.  It is the same as the evaluation of any other expression.  The only thing to look out for is a negative number.
    e.g.
    Evaluate:   yx2z3
    y = 3, x = 4, z = 2
    Solution:  
         (3)(4)2(2)3
         (3)(16)(8) = 384

    Exponent Theorems

    Product Theorem for Exponents

    If m and n are real numbers and x does not equal 0, xm * xn = xm+n

    Quotient Theorem for Exponents

    If m and n are real numbers and x does not equal 0, xm/xn = xm-n = 1/xn-m

    Power Theorem for Exponents

    If m and n are real numbers and x does not equal 0, (xm)n = xmn
    Example:

1.   Simplify:   x2y2x5y3


     Solution:  
     x2x5y2y3
    
     x7y5		   Rearrange the factors so
they are easier to deal with.
Use the Product Theorem to
simplify the expression.
2.   Simplify:   x4
--
x6


     Solution:  
     1
     ----
     x6-4
    
     1
     ---
     x^2		  
Use the Quotient Theorem to
combine the numerator and denominator
into one term in the denominator.
3.   Simplify:   (x-4)-2



     Solution:  
     x8		  
Use the Power Theorem to
multiply the two exponents into one.
    When you come across an expression with many terms, it is easier to deal with that expression when it has been
    simplified by adding like terms.  When doing this with an expression that contains exponents, the variables and their exponents have to be the same.

    Example:
Simplify:   x2yz5 + 2xy2z5 + 3z5x2y - 7y2xz5


     Solution:  
     x2yz5 + 3x2yz5 + 2xy2z5 - 7xy2z5
    
     4x2yz5 - 5xy2z5		   Rearrange the factors so they are more easily identifiable as like terms.
Combine like terms and get this answer.