GRE Algebra | Rules of Exponent

In algebraic expression x^m, x is the base and m is the exponent. For all positive number of x except x=1, if an equation contain x^m = 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