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Rules of Exponent

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This lesson is part of a series of lessons for the quantitative reasoning section of the GRE revised General Test. In this lesson, we will learn:

  • Rules of Exponent
    • Zero Exponents
    • Product Rule
    • Quotient Rule
    • Power Rule
    • Negative Exponent
    • Same Exponent

Rules of Exponent

In the algebraic expression xa, where x is raised to the power a, x is called a base and a is called an exponent.

Here are the basic rules of exponents, where the bases x and y are nonzero real numbers and the exponents a and b are integers.




Zero Exponent

x0 = 1 for x ≠ 0

50 = 1, (−2)0= 1, 00 is undefined

Product Rule

xaxb = xa+b

x2x3 = x2+3 = x5

Quotient Rule

quotient rule

quotient rule

Power Rule

(xa)b = xab

(x2)3 = x2×3 = x6

Negative Exponent

negative exponent

negative exponent

Same Exponent

(xa)(ya) = xya

(2a)(3a) = 6a

The rules above are identities that are used to simplify expressions. Sometimes algebraic expressions look like they can be simplified in similar ways, but in fact they cannot.

Here are several pairs of expressions that are commonly mistaken to be identities.

xayb ≠ (xy)a+b
We can only add the exponents when the bases are the same.

• (xa)bxaxb
Instead, (xa)b = xab and xaxb = xa+b
Example, (x2)3 = x2×3 = x6 and x2x3 = x2+3 = x5

• (x + y)axa + yb
For example, (x + y)2 = x2 + 2xy + y2

• (−x)2 ≠−x2
Instead, (−x)2 = x2

This video shows how to simplify exponential expressions using the power and product rule.

This video shows how to simplify exponential expressions using the power, product and quotient rule.

This video shows how to simplify exponential expressions with negative exponents.

Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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