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 


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


    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 


    (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 


    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 


    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 


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


    (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

My Personal Notes arrow_drop_up

Check out this Author's contributed articles.

If you like GeeksforGeeks and would like to contribute, you can also write an article using or mail your article to See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.

Article Tags :


Please write to us at to report any issue with the above content.