GRE Algebra | Rules of Exponent

In algebraic expression xm, x is the base and m is the exponent. For all positive number of x except x=1, if an equation contain xm = xn then it will be only possible when m = n.

Here are basic rules of Exponents:

  1. If a number raised to the power zero then it should be equal to 1.
    x0 = 1 

    Example:



    20 = 1 
  2. A negative exponent is the same as the reciprocal of the positive exponent.
    x-m = 1/xm 

    Example:

    2-4 = 1/24 = 1/16 
  3. If two powers have the same base then we can multiply the powers. When we multiply two powers we add their exponents.
    (xm) (xn) = xm+n 

    Example:

    (23)(24) = 27 = 128 
  4. If two powers have the same base then we can divide the powers. When we divide powers we subtract their exponents.
    xm/xn = xm-n = 1/xn-m 

    Example:

    34/32 = 32 = 9 
  5. If two powers have different base but same exponent then we multiply the base of powers and exponent will remain same.
    (xm)(ym) = (xy)m 

    Example:

    3242 = 122 = 144 
  6. If base is a fraction then the exponent of the power multiply with numerator and denominator separately.
    (x/y)m = xm/ym 

    Example:

    (2/3)2 = 22/32 = 4/9 
  7. If power has an exponent then both the exponents multiplied.
    (xm)n = xmn 

    Example:

    (32)3 = 36 = 729 

Avoid the common mistakes like below:

  1. xmyn ≠ (xy)m+n 

    Here bases are not same so we cannot add the exponents.

  2. (xm)n ≠ xmxn 

    Here exponents should be multiplied not added according to rule.

  3. (x + y)m ≠ xm + ym 

    Have a look at (x + y)2 = x2 + 2xy + y2



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