An exponent is a number written to the upper right of another number called a base. Exponents are sometimes called powers, as in "2 to the fourth power" or "2 raised to the fourth power." All an exponent is, is a shortcut. It's a way to condense something as unwieldy as 2x2x2x2x2x2x2x2x2x2x2x2x2x2 into a compact "2 to the 14th power," or 2 with a superscript 14 written to its upper right. All numbers are technically bases and all have an understood, invisible exponent of 1 unless stated otherwise.
Expand a base/exponent combination by writing out the base number multiplied by itself. If the base is 3 and the exponent is 5, write this as 3x3x3x3x3, or 243. Condense a string of numbers by counting how many times the number is multiplied by itself, and using the resulting number as the exponent. Reduce numbers by finding the square root, cube root or other root and using that as the base; for example, 9 is composed of three 3's, or 3 cubed or to the third power.
Change a negative exponent to positive by changing the base into a fraction. If you have 6 to the -2 power, change 6 to 1/6, and change the -2 to 2. The exponent stays with the original base number, so you would read this as 1 over 6 squared, or 1 over 6 to the second power, equaling 1/36.
Change any base with a 0 exponent to 1. When dealing with equations in which you have a base with an exponent of 0, you can immediately simplify the base and exponent to the number 1. If the base has an exponent of 1, simplify it by just removing the 1. A base to the first power equals the base x 1.
Simplify an equation with the form a to the b power times a to the c power by condensing both a's into one and placing the two exponents in parentheses to the upper right of the 2. Add the exponents. So, 2 to the third power times 2 to the fourth power is the same as 2 to the 3+4 power. In the original form, you have 8x16=128; in the new form, you have 2 to the seventh power, which also equals 128.
Multiply the exponent outside the parentheses surrounding an equation or value with each exponent within the parentheses, including any assumed exponents of 1. Remember that exponents of 1 aren't always written, but they are assumed if the base number doesn't have another exponent specified. So, (5x3) to the fourth power could also be written as 5 to the fourth power times 3 to the fourth power, or 625x81, or 50,625. An equation like (5 to the fourth power times 3) to the third power could be written (5 to the 4x3 power)x(3 to the 1x3 power). The same procedure applies to fractions within the parentheses; (2/4) to the fourth power equals 2 to the fourth power over 4 to the fourth power.
Reduce super-long numbers ending in zeroes to a smaller decimal multiplied by 10 to x power. This is called scientific notation. If you have a number like 35,000, reduce the number to the smallest possible version that still has one number to the left of the decimal point; in this case, 3.5. Count how many decimal points you had to move to the left in order to get that number -- four in this example -- and use that as an exponent for a base of 10. When you multiply or divide a number by 10, you're really just moving the decimal point over a spot or so, and that's essentially what you're doing here. Saying 10 to the fourth power is the same as saying 10,000, and 3.5x10,000 is 35,000.