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Math Basics: The Laws of Exponents

written by: Keren Perles • edited by: SForsyth • updated: 11/8/2012

There are several laws of exponents that are important to study and understand. This list of exponent rules will give each mathematical law and explain it step by step. For more details, see the specific article in the series that relates to the rule.

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    Rule #1: Multiplying Exponents With the Same Base

    a^m X a^n = a^(m+n)

    To multiply two exponents that have the same base, add the powers. In other words, if you wanted to multiply 3^4 by 3^6, you would get 3^10. This makes a lot of sense. Think about it:

    3^4 = 3 X 3 X 3 X 3

    3^6 = 3 X 3 X 3 X 3 X 3 X 3

    If you multiply them together, you get (3 X 3 X 3 X 3)(3 X 3 X 3 X 3 X 3 X 3), which means there are ten 3s multiplied together, or 3^10.

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    Rule #2: Dividing Exponents With the Same Base

    (a^m)/(a^n) = a^(m-n)

    To divide two exponents that have the same base, subtract the power in the denominator from the power in the numerator. In other words, if you wanted to divide q^4 by q^2, your answer would be q^(4-2) or q^2.

    You can prove this the same way you did with the previous law. If there are four q’s in the numerator (q X q X q X q) and two in the denominator (q X q), two of the q’s will cancel each other out, leaving only two q’s in the numerator.

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    Rule #3: Raising a Product to a Power

    (aXb)^n = a^n X b^n

    To raise a product of several numbers to a power, raise each number to the power. In other words, if you wanted to raise 2q to the third power, you would have to raise the 2 and the q to the third power, so your answer would be 8q^3.

    You can prove this one by writing out (2 X q) X (2 X q) X (2 X q). You can remove the parentheses and combine the 2’s and the q’s like this: 2 X 2 X 2 X q X q X q. Look again – that’s the same thing as 2^3 X q^3 or 8q^3.

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    Rule #4: Raising a Quotient to a Power

    (a/b)^n = (a^n/b^n)

    To raise a quotient of two numbers to a power, raise each number to the power. In other words, if you wanted to raise 2/5 to the third power, you would have to raise the 2 and the 5 to the third power, so your answer would be (2^3)/(5^3) or 8/125.

    You can prove this one by viewing the quotient as a fraction, and multiplying it by itself n times. Try it.

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    Rule #5: Raising an Exponent to an Additional Power

    (a^m)^n = a^(mXn)

    To raise an exponent to an additional power, multiply the two powers. In other words, if you wanted to raise x^2 to the third power, you would multiply the two powers – 2 and 3. This would leave you with the answer x^6. The proof for this law is beyond the scope of this article.

    Just five exponent rules. That’s not so bad, is it? Memorize these five laws of exponents and learn how to apply them. Suddenly, exponents won’t seem so tough at all!

Math Help for Exponents

Looking for math help for exponents? Whether you're a student, parent, or tutor, this series of articles will explain the basics of how to use exponents correctly. Includes rules for adding, subtracting, multiplying, and dividing exponents, as well as how to use negative exponents.
  1. Math Basics: Calculating and Using Exponents
  2. Math Basics: The Laws of Exponents
  3. Adding and Subtracting Exponents
  4. Exponent Study Guide: Multiplying and Dividing Exponents With the Same Bases
  5. Learn Math Basics: About Negative Exponents

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