GRE Algebra | Rules of Exponent

• Last Updated : 24 Apr, 2019

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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