1. 6.1 – Properties of Exponents
Today’s learning goal:
students will be able to use zero and negative exponents
Use properties of exponents
Solve real –life problems involving exponents.

2. Core Concept Example: 4 −2 = 1 4 2 Example: 𝑎 −𝑛 = 1 𝑎 𝑛 , where a≠0
Zero Exponent
For any nonzero number a, a0 = 1. The power of 00 is undefined.
Example: 40 =1
Example: a0 = 1, where a≠0
Negative Exponent
For any integer n and any nonzero number a, a-n is the reciprocal of an.
Example: 4 −2 =
Example: 𝑎 −𝑛 = 1 𝑎 𝑛 , where a≠0

3. Example 1 - Example 2- c. 4𝑥 0 𝑦 −3
Evaluate each expression a = b. (−2) −4 =
1 (−2) 4
= 1 −2∗−2∗−2∗−2 1
Example 2-
= 1 16
Simplify the expression
c. 4𝑥 0 𝑦 −3
= 4𝑥 0 𝑦 3 =4 𝑦 3

4. You Try-
Evaluate each expression 1. (−9) 0 = b. 3 −3 =
1 3 3
= 1 3∗3∗3
= 1 27 1
Simplify the expression using only
positive exponents
c. − − d. 3 −2 𝑥 −5 𝑦 0
= 𝑥 5 = 1 9𝑥 5
= −1∗2 2 =−4

5. Core Concept – Product of Powers Property
To multiply powers with the same base, add their exponents.
4 6 ∗ 4 3 = 4 9
To prove this, lets look at the expanded form of the equation:
4 6 means 4∗4∗4∗4∗4∗4
4 3 means 4∗4∗4
So, 4 6 ∗ 4 3 means 4∗4∗4∗4∗4∗4∗4∗4∗4 which is equivalent to 4 9

6. Example 3 – multiplying with exponents
= 3∗3∗3∗3∗3∗3∗3∗3= 3 8
b. 𝑥 3 ∗ 𝑥 5
= 𝑥 8
c. 𝑥 𝑚 ∗ 𝑥 𝑛
= 𝑥 𝑚+𝑛
d. 2 𝑎 ∗ 2 𝑏
= 2 𝑎+𝑏

7. Core Concept – Quotient of Powers Property
To divide powers with the same base, subtract their exponents
= 4 6−3 = 4 3
To prove this, lets look at the expanded form of the equation:
4∗4∗4∗4∗4∗4 4∗4∗4
= 4 3

8. Example 4 – dividing with exponents
Evaluate each expression a. (−4) 2 (−4) 7 b =
= 1 (−4) 5
=− 1 −4∗−4∗−4∗−4−∗4
=(−4) 2−7
= (−4) −5 =−
5 9−6
= 5 3
=125
c ∗ ∗5 2
= 3∗3∗3∗5∗5∗5 3∗5∗5
=3∗3∗5=45

9. Core Concept – power of a power property
To find a power of a power, multiply the exponents
( 4 6 ) 3 = 4 18
To prove this, lets look at the expanded form of the equation:
( 4 6 ) 3 means 4 6 ∗ 4 6 ∗ 4 6
Which would be 4∗4∗4∗4∗4∗4∗4∗4∗4∗4∗4∗4∗4∗ 4∗4∗4∗4∗4
So, ( 4 6 ) 3 means 4 18

10. Example 4 – Power to a Power
Evaluate each expression a. ( 𝑧 4 ) −3 b. ( 6 −2 ) −1 =
= 1 𝑧 12
=𝑧 4∗−3
= 𝑧 −12
6 −2∗−1
= 6 2
=36
c. ( 𝑤 12 ) 5
= 𝑤 60

11. Core Concept – Power of a product
To find a power of a product, find the power of each factor and multiply
(3∗2) 5 = 3 5 ∗ or (𝑎𝑏) 𝑚 = 𝑎 𝑚 ∗ 𝑏 𝑚 = 𝑎 𝑚 𝑏 𝑚
Example 5
Evaluate each expression a. (−1.5𝑦) 2 b. ( 𝑎 −10 ) 3 =
=(−1.5) 2 ∗ 𝑦 2
= 2.25𝑦 2
𝑎 3 (−10) 3
=− 𝑎
c. ( 3𝑑 2 ) 4
= 𝑑
= 81 𝑑 4 16

12. Example 5 – Solving a real-life problem
A jellyfish emits about 1.25 x 10 8 particles of light, or photons, in 6.25 x 10 −4 second. How many photons does the jellyfish emit each second? Write you answer in scientific notation and in standard form. a 𝑥 𝑥 10 −4
= 𝑥 −4
=0.2 𝑥
=2 𝑥
The jellyfish emits 200,000,000,000 photons per second.

13. You try!
It takes the Sun about 2.3 x years to orbit the center of the Milky Way. It takes Pluto about 2.5 x years to orbit the Sun. How many times does Pluto orbit the Sun while the Sun completes one orbit around the center of the Milky Way? Write you answer in scientific notation.
2.3 𝑥 𝑥 10 2
= 𝑥
=0.92 𝑥 10 6
=9.2 𝑥 10 5