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Exponent Rules

by Ron Kurtus (14 November 2015)

When you perform algebraic operations on exponential expressions, there are several exponent rules that you can follow. These rules are the product rule, quotient rule, and power rule.

Questions you may have include:

This lesson will answer those questions.



Exponent product rule

When you multiply two exponential expressions with the same base, you simply add the exponents:

x4 * x2 = x4 + 2 = x6

23 * 25 = 23 + 5 = 28

ya * yb = ya + b

When you multiply two exponential expressions with the same exponent, the product of the bases is raised to that exponential power:

xa * ya = (xy)a

33 * 23 = (3 * 2)3 = 63

Exponent quotient rule

When you divide two exponential expressions with the same base, you simply subtract the denominator (lower) exponent from the numerator (upper) exponent:

x5/x2 = x5 − 2 = x3

27/25 = 27 − 5 = 22

ya/yb = ya − b

If the denominator (lower) exponent is greater than the numerator (upper) exponent, the result will have a negative exponent:

y3/y5 = y3 − 5 = y−2 = 1/y2

36/38 = 36 − 8 = 3−2 = 1/32 = 1/9

When you divide two exponential expressions with the same exponent, the quotient of the bases is raised to that exponential power:

xa/ ya = (x/y)a

33/23 = (3/2)3

Exponent power rule

When you raise an exponential expression to a power, you simply multiply the exponents:

(ya)b = yab

(72)3 = 76

Proof:

(72)3 = (72)(72)(72) = 72 + 2 + 2 = 76

Summary

When you perform algebraic operations on exponential expressions, there are several exponent rules that you can follow. These rules are the product rule where you add the exponents, the quotient rule where you subtract the denominator exponent from the numerator exponent, and the power rule where you multiply the exponents.


Learn the rules and follow them


Resources and references

Ron Kurtus' Credentials

Websites

Exponents: Basic Rules - PurpleMath.com

Exponent Rules - RapidTables.com

Laws of Exponents - MathisFun.com

Exponents Calculator - CalculatorSoup.com

Algebra Resources

Books

Top-rated books on Algebra


Questions and comments

If you have questions, comments, or opinions on this subject, send an email with your feedback. I will try to get back to you as soon as possible.


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