# 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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