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 x^a, 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.

Name	Rule	Example
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
Power Rule	(xa)b = xab	(x2)3 = x2×3 = x6
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.

• x^ay^b ≠ (xy)^a+b
We can only add the exponents when the bases are the same.

• (x^a)^b ≠x^ax^b
Instead, (x^a)^b = x^ab and x^ax^b = x^a+b
Example, (x²)³ = x^2×3 = x⁶ and x²x³ = x²⁺³ = x⁵

• (x + y)^a ≠x^a + y^b
For example, (x + y)² = x² + 2xy + y²

• (−x)² ≠−x²
Instead, (−x)² = x²

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

• Show Step-by-step Solutions

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

• Show Step-by-step Solutions

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

• Show Step-by-step Solutions

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