Exponents are used to shorten expressions by simplifying like terms that are multiplied. For example, the expression z z z could be simplified using exponents because there are like terms. Since there are three Zs being multiplied, it would be simplified to z ^ 3. Z is the base and three is the exponent.
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Add the exponents if the terms are being multiplied. X ^3 * X ^ 2 would be simplified to X ^ 5 because 2 + 3 = 5.

Subtract the exponents if the terms are being divided. C ^ 9 / C ^ 2 would be simplified to C ^ 7 because 9  2 = 7.

Multiply the exponents if the term is being raised to a power. (Y^3)^4 would be simplified to Y ^ 12 because 3 * 4 = 12.
Tips & Warnings
 Only exponents with common bases can be combined. For example, x ^ 3 * y ^ 3 cannot be simplified.
References
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