Calculations involving exponents are crucial in algebra and are a major feature of SAT/ACT math. Seven basic rules and two additional corollaries govern exponentiation.
It’s important to understand these principles well and master their use through practice and application.
Below are the laws governing exponents you'll need to know and follow.
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Basic Exponent Rules (A ≠ 0, B ≠ 0)
Product of Equal-Base Powers: A^m*A^n = A^(m+n)
c: z^3*z^4 = z^7.
Quotient of Equal-Base Powers: A^m/A^n = A^(m–n).
For example: x^-3/x^5 = x^-8.
Power of a Power: (A^m)^n = A^(mn)
For example: (y^3)^4 = y^12.
Power of a Product: (A*B)^n = (A^n)(B^n)
For example: (x^2*y)^3 = (x^6)(y^3).
Power of a Quotient: (A/B)^n = [(A)^n]/[(B)^n]
For example: (x^7.5/y^-2)^2 = [x^15]/[ y^-4].
Zero Powers: A^0 = 1 (A ≠ 0)
For example: (2z–1)^0 = 1 (z ≠ 1/2).
Negative Powers: A^-n = 1/(A^n)
For example: x^-3 = 1/(x^3).
Additional Corollaries
Quotient of Negative Powers: A^-m/B^-n = B^n/A^m
For example: y^-1/z^4 = z^-4/y^1.
(Changing positions of the lower and/or upper powers changes the signs on those exponents)
Negative Power of a Quotient: (A/B)^-n = (B/A)^n
For example: (x^-1/y^8)^-4 = (y^8/x^-1)^4.
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For practice, search Google for “exponent rules worksheet,” pick a worksheet that provides answers, complete the worksheet, analyze any mistakes, and redo it until you can complete that worksheet with no errors. Then repeat, with additional worksheets, as needed, until you’ve mastered this material.
Below are the laws governing exponents you'll need to know and follow.
-
Basic Exponent Rules (A ≠ 0, B ≠ 0)
Product of Equal-Base Powers: A^m*A^n = A^(m+n)
c: z^3*z^4 = z^7.
Quotient of Equal-Base Powers: A^m/A^n = A^(m–n).
For example: x^-3/x^5 = x^-8.
Power of a Power: (A^m)^n = A^(mn)
For example: (y^3)^4 = y^12.
Power of a Product: (A*B)^n = (A^n)(B^n)
For example: (x^2*y)^3 = (x^6)(y^3).
Power of a Quotient: (A/B)^n = [(A)^n]/[(B)^n]
For example: (x^7.5/y^-2)^2 = [x^15]/[ y^-4].
Zero Powers: A^0 = 1 (A ≠ 0)
For example: (2z–1)^0 = 1 (z ≠ 1/2).
Negative Powers: A^-n = 1/(A^n)
For example: x^-3 = 1/(x^3).
Additional Corollaries
Quotient of Negative Powers: A^-m/B^-n = B^n/A^m
For example: y^-1/z^4 = z^-4/y^1.
(Changing positions of the lower and/or upper powers changes the signs on those exponents)
Negative Power of a Quotient: (A/B)^-n = (B/A)^n
For example: (x^-1/y^8)^-4 = (y^8/x^-1)^4.
-
For practice, search Google for “exponent rules worksheet,” pick a worksheet that provides answers, complete the worksheet, analyze any mistakes, and redo it until you can complete that worksheet with no errors. Then repeat, with additional worksheets, as needed, until you’ve mastered this material.
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