Zero and Negative Exponents

Examples, solutions, and videos to help Grade 8 students learn about zero and negative exponents.

New York State Common Core Math Grade 8, Module 1, Lesson 4 and lesson 5.

Download Worksheet for Common Core Grade 8, Module 1, Lesson 4

Download Worksheet for Common Core Grade 8, Module 1, Lesson 5

Lesson 4 Student Outcomes

Students know that a number raised to the zeroth power is equal to one.

Students recognize the need for the definition to preserve the properties of exponents.

Lesson 4 Opening Exercise

For any numbers x, y and any positive integers m, n, the following holds:

x^m • xⁿ = x^m+n
(x^m)ⁿ= x^mn
(xy)ⁿ = xⁿyⁿ

Definition: For any positive number x, we define x⁰ = 1.

Examples

1. Simplify the following expression as much as possible.
9¹⁵• 9^-10 • 9^-5 =

4¹⁰/4¹⁰ • 7⁰ =

Lesson 5 Student Outcomes

Students know the definition of a number raised to a negative exponent.

Students simplify and write equivalent expressions that contain negative exponents.

Lesson 5 Opening Exercise

For any numbers x, y and any positive integers m, n, the following holds:

x^m • xⁿ = x^m+n
(x^m)ⁿ= x^mn
(xy)ⁿ = xⁿyⁿ

Definition: For any positive number x and for any positive integer n, we define x^-n = 1/xⁿ.
Note that this definition of negative exponents says x^-1 is just the reciprocal 1/x of x.

x^m ÷ xⁿ = x^m-n
(x/y)ⁿ = xⁿ/yⁿ

Examples

2^-1 =
3^-2 =
(4/5)^-3 =
5^-2 =
1/8⁹ =
3 • 2^-4=

Let x and y be non-zero numbers
x^-3 =
1/x⁹ =
xy ^-4 =

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