1. Section 7.3 Rational Exponents
Algebra 1

2. Learning Targets Define a rational exponent Define radical form
Define and apply “ n th ” root
Evaluate “ 𝑛 𝑡ℎ ” root expressions
Define and apply Power Property of Equality
Solve exponential equations

3. Recall: Exponent Definition
3 4
=3∙3∙3∙3=81
Remember, this is saying that I want 4 pieces of the base multiplied together

4. Multiplicative VS Additive Half
12 = 6+6
18=9+9
16=8+8
Multiplicative Half:
16=4∙4
36 = 6∙6
25= 5∙5
Thus, the additive half of 16 is 8 and the multiplicative half of 16 is 4.

5. Multiplicative VS Additive Thirds
9= 3+3+3
12 = 4+4+4
30=
Multiplicative Third:
8= 2 ∙2 ∙2
27= 3 ∙3 ∙3
125= 5∙5∙5
Thus, the additive third of 30 is 10 and the multiplicative third of 125 is 5.

6. Explore: Exponent Definition
What if I have something like…
This is saying that I know 81 is made of 4 equal pieces and I only want 1 of those pieces. Or I want the multiplicative fourth of 81
Thus, 81=3∙3∙3∙3 and one of the four pieces is just 3.
So, =3

7. Concept 1: Basic Rational Exponent
The most common rational exponent is 𝑥 = 𝑥
𝑥 is also known as the square root. This representation is also known as radical form.
𝑥 is asking for the multiplicative half of a number 𝑥

8. Concept 1: Basic Rational Exponent
Practice 1:
What is ?
16=4∙4
Thus, =4
Practice 2:
Find 100
100=10∙10
Thus, =10

9. Concept 2: “ 𝒏 𝒕𝒉 ” Roots If 𝑎 𝑛 =𝑏, then 𝑏 1 𝑛 = 𝑛 𝑏 =𝑎
Ex: 2 4 =16, then = 4 16 =2
4 16 = This is saying, I know 16 has 4 equal multiplicative pieces. It’s then asking, what is 1 of those 4 pieces.
2∙2∙2∙2=16
Thus, =2

10. Concept 2: “ 𝒏 𝒕𝒉 ” Roots Practice 1: Practice 2: What is 27 1 3 ?
27=3∙3∙3
Thus, =3
Practice 2:
Find
32=2∙2∙2∙2∙2
Thus, =2

11. Concept 2: “ 𝒏 𝒕𝒉 ” Roots Practice 3: Practice 4: What is 64 1 3 ?
64=4∙4∙4
Thus, =4
Practice 4:
Find
125=5∙5∙5
Thus, =5

12. Concept 3: Advanced “ 𝒏 𝒕𝒉 ” Roots
𝑏 𝑚 𝑛 = 𝑛 𝑏 𝑚
Ex: = =8
= This is saying, I know 16 has 4 equal multiplicative pieces. It’s then asking, what is 3 of those 4 pieces.
2∙2∙2∙2=16
Thus, =8

13. Concept 3: Advanced “ 𝒏 𝒕𝒉 ” Roots
Practice 1:
What is ?
27=3∙3∙3
Thus, =9
Practice 2:
Find
36=6∙6
Thus, =216

14. Concept 3: Advanced “ 𝒏 𝒕𝒉 ” Roots
Practice 3:
What is ?
64=4∙4∙4
Thus, =16
Practice 4:
Find
32=2∙2∙2∙2∙2
Thus, =4

15. Concept 4: Solving Exponential Equations
Power Property of Equality
For any real number 𝑏>0 and 𝑏≠1, then 𝑏 𝑥 = 𝑏 𝑦 if and only if 𝑥=𝑦.
Example 1: If 5 𝑥 = 5 3 , then 𝑥=3
Example 2: If 2 𝑥+1 = 2 7 , then 𝑥+1=7

16. Concept 4: Solving Exponential Equations
Practice 2:
Solve 25 𝑥−1 =5
5 2 𝑥−1 = 5 1
2 𝑥−1 =1
2𝑥−2=1
Thus, 𝑥= 3 2
Practice 1:
Solve 6 𝑥 =216
6 𝑥 = 6 3
Thus, 𝑥=3

17. Concept 4: Solving Exponential Equations
Practice 4:
Solve 12 2𝑥+3 =144
12 2𝑥+3 = 12 2
2𝑥+3=2
Thus, 𝑥=− 1 2
Practice 3:
Solve 5 𝑥 =125
5 𝑥 = 5 3
Thus, 𝑥=3

18. Exit Ticket for Feedback
1. Solve 4 2𝑥−1 = 2 3
2. Find