1. 4.4 Fractional Exponents and Radicals

2. Construct Understanding

3. Construct Understanding

4. REMEMBER Grade 9?
𝒂 𝒎
BASE

5. REMEMBER Grade 9?
𝒂 𝒎
EXPONENT

6. 𝒂 𝒎 • 𝒂 𝒏 = 𝒂 𝒎+𝒏 REMEMBER Grade 9?
𝒂 𝒎 • 𝒂 𝒏 = 𝒂 𝒎+𝒏
We can further use it to calculate fractional exponents with numerator 1…

7. WHAT IS A FRACTIONAL EXPONENT?
𝒂 x y

8. A FRACTIONAL EXPONENT with a numerator 1
𝒂 1 y

9. Without a calculator, Calculate
𝟓 𝟏 𝟐 • 𝟓 𝟏 𝟐 = 𝟓 𝟏 𝟐 + 𝟏 𝟐 = 𝟓 𝟐 𝟐 = 𝟓
𝟓 • 𝟓 = 𝟐𝟓 = 𝟓
What do you notice?

10. 𝟓 𝟏 𝟐 and 𝟓 equivalent expressions
𝟓 𝟏 𝟐 = 𝟓
𝟓 𝟏 𝟐 • 𝟓 𝟏 𝟐 = 𝟓 𝟏 𝟐 + 𝟏 𝟐 = 𝟓 𝟐 𝟐 = 𝟓
𝟓 • 𝟓 = 𝟐𝟓 = 𝟓
𝟓 𝟏 𝟐 and 𝟓 equivalent expressions

11. 𝟓 𝟏 𝟐 = 𝟓
𝟓 𝟏 𝟐 • 𝟓 𝟏 𝟐 = 𝟓 𝟏 𝟐 + 𝟏 𝟐 = 𝟓 𝟐 𝟐 = 𝟓
𝟓 • 𝟓 = 𝟐𝟓 = 𝟓
Similarly,

12. Without a calculator, Calculate
𝟓 𝟏 𝟑 · 𝟓 𝟏 𝟑 · 𝟓 𝟏 𝟑 = 𝟓 𝟏 𝟑 + 𝟏 𝟑 + 𝟏 𝟑 = 𝟓 𝟑 𝟑 = 𝟓
𝟑 𝟓 · 𝟑 𝟓 · 𝟑 𝟓 = 𝟑 𝟏𝟐𝟓 = 𝟓
What do you notice?

13. 𝟓 𝟏 𝟑 and 𝟑 𝟓 equivalent expressions
𝟓 𝟏 𝟑 = 𝟑 𝟓
𝟓 𝟏 𝟑 · 𝟓 𝟏 𝟑 · 𝟓 𝟏 𝟑 = 𝟓 𝟏 𝟑 + 𝟏 𝟑 + 𝟏 𝟑 = 𝟓 𝟑 𝟑 = 𝟓
𝟑 𝟓 · 𝟑 𝟓 · 𝟑 𝟓 = 𝟑 𝟏𝟐𝟓 = 𝟓
𝟓 𝟏 𝟑 and 𝟑 𝟓 equivalent expressions

14. 𝟓 𝟏 𝟐 and 𝟓 equivalent expressions
𝟓 𝟏 𝟐 = 𝟓
𝟓 𝟏 𝟐 • 𝟓 𝟏 𝟐 = 𝟓 𝟏 𝟐 + 𝟏 𝟐 = 𝟓 𝟐 𝟐 = 𝟓
𝟓 • 𝟓 = 𝟐𝟓 = 𝟓
𝟓 𝟏 𝟐 and 𝟓 equivalent expressions

15. This suggests
𝟓 𝟏 𝟐 = 𝟓
𝟓 𝟏 𝟑 = 𝟑 𝟓
𝒙 𝟏 𝒏 = 𝒏 𝒙

16. Powers with Rational Exponents
𝟓 𝟏 𝟐 = 𝟓
𝟓 𝟏 𝟑 = 𝟑 𝟓
𝒙 𝟏 𝒏 = 𝒏 𝒙
When n is a natural number and x is a rational number,

17. Evaluate each power without using a calculator
𝟐𝟕 𝟏 𝟑 = 3 27 = 3 𝟎.𝟒𝟗 𝟏 𝟐 = 0.49 = 0.7 ( 𝟒 𝟗 ) 𝟏 𝟐 = 𝟒 𝟗 = 𝟐 𝟑

18. POWERPOINT PRACTICE PROBLEM Evaluate each power without using a calculator

19. The numerator in the exponent IS NOT 1?
What if….
The numerator in the exponent IS NOT 1?
𝟖 𝟏 𝟑
𝟖 𝟐 𝟑
𝒙 𝟏 𝒏 = 𝒏 𝒙 ??

20. RECALL..
(𝒂 𝒎 ) 𝒏 = 𝒂 𝒎 • 𝒏
So, for example,

21. 𝟖 𝟐 𝟑 = 𝟖 𝟏 𝟑 · 𝟐
(𝒂 𝒎 ) 𝒏 = 𝒂 𝒎 • 𝒏
𝒂 𝒎 • 𝒏 = (𝒂 𝒎 ) 𝒏
= ( 𝟖 𝟏 𝟑 ) 𝟐
But, this is also true…
= ( 𝟑 𝟖 ) 𝟐
= (𝟐) 𝟐
= 4

22. 𝟖 𝟐 𝟑 = 𝟖 𝟐 · 𝟏 𝟑
(𝒂 𝒎 ) 𝒏 = 𝒂 𝒎 • 𝒏
𝒂 𝒎 • 𝒏 = (𝒂 𝒎 ) 𝒏
= ( 𝟖 𝟐 ) 𝟏 𝟑
= 𝟑 𝟖 𝟐
But, this is also true…
= 𝟑 𝟔𝟒
𝒙 𝟏 𝒏 = 𝒏 𝒙
= 4

23. Powers with Rational Exponents
When m and n are natural numbers and x is a rational number,
𝒙 𝒎 𝒏 = ( 𝒙 𝟏 𝒏 ) 𝒎
= ( 𝒏 𝒙 ) 𝒎
𝒙 𝒎 𝒏 = ( 𝒙 𝒎 ) 𝟏 𝒏
= 𝒏 𝒙 𝒎
AND

24. Write 𝟒𝟎 𝟐 𝟑 in radical form in 2 ways
Write 𝟑 𝟓 and ( 𝟑 𝟐𝟓 ) 𝟐 in exponent form.
𝟒𝟎 𝟐 𝟑 = ( 𝟑 𝟒𝟎 )² and 𝟑 𝟒𝟎²
𝟑 𝟓 = 𝟑 𝟓 𝟐
( 𝟑 𝟐𝟓 ) 𝟐 = 𝟐𝟓 𝟐 𝟑
𝒙 𝒎 𝒏 = ( 𝒙 𝟏 𝒏 ) 𝒎
= ( 𝒏 𝒙 ) 𝒎
𝒙 𝒎 𝒏 = ( 𝒙 𝒎 ) 𝟏 𝒏
= 𝒏 𝒙 𝒎

25. POWERPOINT PRACTICE PROBLEM a) Write 𝟐𝟔 𝟐 𝟓 in radical form in 2 ways b) Write 𝟔 𝟓 and ( 𝟒 𝟏𝟗 ) 𝟑 in exponent form.

26. A husky with a body mass of 27 kg
Biologists use the formula b = 0.01 𝒎 𝟐 𝟑 to estimate the brain mass , b kilograms, of a mammal with body mass m kilograms. Estimate the brain mass each animal
A husky with a body mass of 27 kg
A polar bear with a body mass of 200g

27. Biologists use the formula b = 0
Biologists use the formula b = 0.01 𝒎 𝟐 𝟑 to estimate the brain mass , b kilograms, of a mammal with body mass m kilograms. Estimate the brain mass each animal
A husky with a body mass of 27 kg
A polar bear with a body mass of 200g
Substitute: m = 27
b = 0.01 (𝟐𝟕) 𝟐 𝟑
b = 0.01 (∛𝟐𝟕) 𝟐
b = 0.01 (𝟑) 𝟐
b = 0.09 kg
The brain mass of the husky is approximately 0.09 kg.

28. b = 0.01 (𝟐𝟎𝟎) 𝟐 𝟑 USE a CALCULATOR! Substitute: m = 200
Biologists use the formula b = 0.01 𝒎 𝟐 𝟑 to estimate the brain mass , b kilograms, of a mammal with body mass m kilograms. Estimate the brain mass each animal
A husky with a body mass of 27 kg
A polar bear with a body mass of 200g
Substitute: m = 200
b = 0.01 (𝟐𝟎𝟎) 𝟐 𝟑
USE a CALCULATOR!
The brain mass of the polar bear is approximately 0.34 kg.

29. POWERPOINT PRACTICE PROBLEM Use the formula b = 0
POWERPOINT PRACTICE PROBLEM Use the formula b = 0.01 𝒎 𝟐 𝟑 to estimate the brain mass of each animal. A moose with a body mass of 512 kg A cat with a body mass of 5 kg

30. Section 4.4 HOMEWORK
PAGES: 227 – 228
PROBLEMS:

31. QUIZ
Sections 4.1 – 4.3
Friday, July 25