1. 0.4 Nth Roots and Real Exponents

2. Bell Ringer
Which of the following lists all the roots of

3. 0.4 Nth Roots and Real Exponents
Yesterday finished up our study on Quadratics.
Today we will begin our study on exponential functions by investigating the properties of real exponents. We will:
Find nth roots
Simplify using absolute value
Use the properties of exponents
Evaluate and simplify expressions containing rational exponents.
Solve equations containing rational exponents
Tomorrow we will be solving systems of linear equations .

4. Real nth roots of Real Numbers

5. Simplify = = = = = = = = = = Perfect Square Factor * Other Factor
LEAVE IN RADICAL FORM = = = = = =

6. GROWING A
FACTOR TREE

7. 180 18 10 Can we grow a tree of the factors of 180?
Or You might notice that 180 has a ZERO in its ONES PLACE which means it is a multiple of 10.
SO… 10 x  = 180
10 x 18 = 180
Or You might notice that 180 has a ZERO in its ONES PLACE which means it is a multiple of 10.
SO… 10 x  = 180
180
You might see that 180 is an EVEN NUMBER and that means that 2 is a factor…
2 x  = 180 ?
Can you think of one FACTOR PAIR for 180 ?
This should be two numbers that multiply together to give the Product 180.
Can we grow a tree of the factors of 180? 10 18

8. You have to find FACTOR PAIRS for 10 and 18
180
NOW
You have to find FACTOR PAIRS for 10 and 18
We “grow” this “tree” downwards since that is how we write in English (and we can’t be sure how big it will be - we could run out of paper if we grew upwards). 10 18

9. 180
Find factors for 10 & 18 18 10
2 x 5 = 10
6 x 3 = 18 5 2 6 3

10. 5 10 2 6
180 18 3
Since 2 and 3 and 5 are PRIME NUMBERS they do not grow “new branches”. They just grow down alone.
Since 6 is NOT a prime number - it is a COMPOSITE NUMBER - it still has factors. Since it is an EVEN NUMBER we see that:
6 = 2 x 
ARE WE
DONE
???
2*3=6 2 5 3 2 3

11. … and if we flip it over we can see why it is called a tree2 3 5 10 6
180 18
… and if we flip it over we can see why it is called a tree

12. Examples
Simplify each expression: a) b) c)

13. Simplify the expression:
Your Turn
Simplify the expression:

14. Simplify Using Absolute Value
Simplify Use division. Divide the index number into the exponent. The remainder goes under the radical. Any time you take out an odd exponent with an even index number, absolute value symbols are required.

15. Simplify.
Examples

16. Simplifying Using Absolute Value
Your Turn

17. Motivating the Lesson Property used Use the properties of exponents to
find a number x such that
Property used

18. Simplify each expression.
Examples

19. Simplify each expression.
Your Turn
Simplify each expression.

20. State whether represent the same quantity. Explain.
Think about it…….

21. Simplify each expression.
Examples b. c. a.

22. Simplify the expression.
Your Turn

23. Simplify each expression.
Examples c. a. b.

24. Simplify the expression.
Your Turn
Simplify the expression.

25. Simplify the expression.
Examples

26. Simplify the expression.
Your Turn
Simplify the expression.

27. Examples a. Express using rational exponents.
b. Express using a radical.

28. If au = av, then u = v Example
This says that if we have exponential functions in equations and we can write both sides of the equation using the same base, we know the exponents are equal.
Example
The left hand side is 2 to the something. Can we re-write the right hand side as 2 to the something?
Now we use the property above. The bases are both 2 so the exponents must be equal.
We did not cancel the 2’s, We just used the property and equated the exponents.
You could solve this for x now.

29. The left hand side is 4 to the something but the right hand side can’t be written as 4 to the something (using integer exponents)
Let’s try one more:
We could however re-write both the left and right hand sides as 2 to the something.
So now that each side is written with the same base we know the exponents must be equal.
Check:

30. Your Turn
Solve

31. HOT Question If 5-2 is raised to the third power,
Determine whether the statement makes sense or
does not make sense, and explain your reasoning.
If 5-2 is raised to the third power,
the result is between 0 and 1.

32. Wrap-up
How do you simplify a radical?
Which is correct:

33. Practice makes perfect!
0.4
Practice
Practice makes perfect!