# 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.

 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.

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.

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