Download Discovering Zero and Negative Exponents: Understanding Reciprocal Powers and more Lecture notes Calculus in PDF only on Docsity!
Discovering Zero and Negative Exponents
Directions: Complete the table with a partner. Show both ways. Try to follow patterns and use what you know about laws of exponents.
Name:
Key
Evaluate the power. Take the previous answer and divide by 2. (Decompose and find equivalent forms of 1)
Show using laws of exponents. What happens?
2!^ = 16
2!^
2!^
The exponent decreases by 1.
2!^ = 8
2!^
2!^
The exponent decreases by 1.
2!^ = 4
2!^
2!^
The exponent decreases by 1.
2!^ = 2 2!
2!^
2!^
The exponent decreases by 1 and we see that 2!^ = 1.
2!^ = 1 2!
2!^
2!^
The exponent decreases by 1 and we see that 2 !!^ =
! !.
2 !!^ =
2!^
2!^
The exponent decreases by 1 and we see that 2 !!^ =! !.
2 !!^ =
2!^
2!^
The exponent decreases by 1 and we see that 2 !!^ =
! !.
You Try! Complete the table below using what you now know about a zero exponent or a negative exponent.
NOTE: (Have these pre-‐made on chart paper so that students may come up and fill in solutions as they are found.)
5!^10!
5!^ = !"# 10!^ = !!!!! Make an observation : Describe
some of the patterns you see going down the columns.
repeatedly by 5 or 10.
exponents are
reciprocals.
negative despite the
negative exponents.
5!^ = !"# 10!^ = !"""
5!^ = !" 10!^ = !""
5!^ =! 10!^ = !"
5!^ =! 10!^ =!
5 !!^ =
!
!"
5 !!^ =
!
!""
5 !!^ =
!
!"""
Negative Exponents
A) Compare the two powers: 2!^ and 2 !!. How would you describe the two answers? (CHORAL RESPONSE) “They are reciprocals.”
B) Compare the two powers: 2!^ and 2 !!. How would you describe the two answers? (CHORAL RESPONSE) “They are reciprocals.”
Make a prediction : Based on the last questions, if 2 !!^ =! !, and 2 !!^ =! !, and 2!^ = 8 then 2 !!^ =
! !
Therefore, we have determined that a negative exponent is like a command. It tells us to:
Find the reciprocal of the base…before we evaluate the power.
Note: Negative exponents do not yield negative answers. (Unless, in some cases, when the base itself is negative.)
For every non-‐zero number! and integer, !, !!!^ =
!
!!^
.
Zero Exponent
C) What did you notice when dividing 2
1
21
? (Hint, you should have found two answers, 2!^ and 1. )
We found that!
! !!^ =^!^ and^
!! !!^ =^!
! (^) therefore, !! (^) = !.
D) Make a prediction: What is the answer to a problem like
!!^
or
The answer would be 1 or !!^ =! and !!^ =!.
E) Therefore, we have determined that an exponent of zero always gives an answer of: (CHORAL RESPONSE) One!
For every non-‐zero number !, !!^ = 1.
Raising any Base to a Zero Power
Simplify !!!!!^!^ Power of a Product Rule
We may apply two rules of exponents. (^) !!!!!^!^ =!!^ !!^!^ !!^!^ Power of a Power Rule
= !!!!∙!!!∙! We may now apply the Zero Exponent Rule. (^) = !!!!!!
= 1
Directions: Simplify using exponent rules. (Debrief using THINK-‐PAIR-‐SHARE)
Example:
!! !! = 2 !!! = 2! 1
You try!
!! !! = !!!! = !! = 1
You try!
!!! !!! = 11 !!! = 11! = 1
Example: !! = 1 You try!^
! !
!
You try! !"!!^
!
=!!^!!^ !!^!
Raising a Quotient to a Negative Power
Simplify
We should first find
the reciprocal of! !.
Method # ! !
=
= (
)!
=
=
=
Notice : When we have found the reciprocal, the exponent becomes positive.
Why? We have completed the action of taking the reciprocal.
In Method #2 we evaluated the power first and took the reciprocal at the end of the problem.
Method # ! !
=
= (
)!!
=
=
=
After this, we evaluate the power.
Example 3 : Evaluate using laws of exponents and negative and zero exponents.
(Remember, a negative exponent indicates that we should find the reciprocal and THEN evaluate the power.)
NOTE: Ask students to show multiple methods on these problems if there is time. Display to class as well.
Example:
! !
!!
!!∙!
!!!
!
You try!
! !!
!!
!!∙!
!!!
!
!!^ ∙ !!^ ∙ !!^ ∙ !!^ ∙ !!
!!^!
You try!
!! !"!!!
!!
!!∙!
!!!
!
12 ∙ 12 ∙ !!^ ∙ !!^ ∙! ∙!
