Integrated Math II Name: ________________________________________ Period: ______ Date: _______________

1

6.1.4 Radicals and Fractional Exponents Do Now 1. A radical is an expression that has a ______________________. For example √100 is a radical. 2. n the term 4!, 4 is the __________ and 3 is the ______________________. 3. The laws of exponents state

a. 𝑥! ∙ 𝑥! = 𝑥!!! then 100! ∙ 100!" = ________ b. 𝑥! ! = 𝑥!" then (7

!!)! = ___________________

c. 𝑥!! = !!!

then 10!! = ___________ d. 𝑥! = 1 then 3! = ____ e. !!

!!= 𝑥!!! then !

!

!!= _____________________

4. √9 = ___, √100 = ___ and √49 = ___ 5. The inverse operation of square rooting is ______________. 6. In the equation 3 ∙ 4 = 12, ___ and ___ are factors because they are numbers we can ________________

together to get another number. Notes

Radical Form of Expressions with Fractional Exponents

For all 𝑥 ≠ 0 and 𝑛 ≠ 0, 𝑥!! = 𝑥!

!! = 𝑥!! or 𝑥

!! = 𝑥

!!

!= 𝑥! !

. Example 1: 5

!!

= 5!!!

= 5!! = 5

Example 2: 3!!

= 3!!!

= 3!! = 9!

Example 3: 16!!

= 16!!

!

= 16! !

= 2 ! =8

Problems In a previous course you learned the laws of exponents. But what if the exponent is a fraction? And how are exponents related to the radicals you have been using in the previous lessons? In this lesson, you will investigate the relationship between radical expressions and fractional exponents. 6-39. Now that you have many tools to rewrite expressions with exponents, use these tools together to rewrite each of the expressions below in two different ways. For example, √25 = (25)1/2 = 25/2, since taking the square root of a number is the same as raising that number to the one-half power. a. (√3)4

= (31/2)4

= 32

= 9

b. 97/2

= (32)7/2

= 37

= (91/2)7 ;

= 9!

c. 2!!

= (25)1/3

= 25/3

Integrated Math II Name: ________________________________________ Period: ______ Date: _______________

2

6-36. FRACTIONAL EXPONENTS. What happens when an exponent is a fraction? Consider this as you answer the questions below. a. Calculate 91/2 using your calculator. What is the result? Also use your calculator to calculate 491/2 and

1001/2. What effect does having ½ in the exponent appear to have? 91/2 = 3. The effect of having ½ in the exponent is the result equals the square root of the base. b. Based on your observation in part (a), predict the values of 41/2 and (71/2)2. Then confirm your predictions

using your calculator. 41/2 = √4 = 2 and (71/2)2 = (√7)2 = 7 6-37. Danielle wants to understand why 91/2 is the same as √9. Since exponents represent repeated multiplication, Danielle decided to rewrite the number 9 as 3 · 3. She then reasoned that 91/2 is asking for one of the two repeated factors with a product of 9. a. Using Danielle’s logic, what is the value of 161/2? Confirm your answer using your calculator. The value of 161/2 = √16 = 4 b. What is the value of 81/3? 1251/3? How can you use the same reasoning to determine these values? Confirm

your answers using your calculator.

The value of 81/3 is 2 and 1251/3 is 5. You can rewrite each number as the product of three repeated factors and then take one of the factors that is 8 = 2 ∙ 2 ∙ 2 so 81/3 = 2 and 125 = 5 ∙ 5 ∙ 5 so 1251/3 = 5.

c. What about 272/3? 323/5? 253/2? Use your calculator to determine each of these values. Then apply Danielle’s logic to make sense of what each of these expressions mean

27 = 3·3·3, so 272/3 is 3·3 = 9; 32 = 2·2·2·2·2, so 323/5 = 2·2·2 = 8; 25 = 5·5, so 253/2 = 5·5·5 = 125

d. Another name for x1/3 is “cube root”. This can be written 𝑥! . What would be the notation for x1/5? What should it be called?

The notation for x1/5 could be 𝑥! , which should be called the fifth root.

Integrated Math II Name: ________________________________________ Period: ______ Date: _______________

3

6-38. EXPONENT LAW FOR RATIONAL EXPONENTS. Addison’s teacher challenges his team to use algebra and the properties of exponents to prove that √3 = 31/2. Addison says, “Well, let’s assume there is some exponent that will give us the square root.” a. Addison writes the equation √3 = 3x. Imani says, “I think we should start by getting rid of the square

root.” What operation will “undo” the square root? The operation the will undo a square root is squaring. b. Starting with Imani’s idea, solve the equation √3 = 3x for x. What does your answer mean? Work:

Response: The exponent ½ gives the square root.

c. Can you extend the argument to other radical expressions? Use a similar technique to demonstrate that

17! = 171/3. 6-40. Match each expression below on the left (letters (a) through (h)) with an equivalent expression on the right (numbers 1 through 8). Assume x > 0.

Integrated Math II Name: ________________________________________ Period: ______ Date: _______________

4

Notes Rationalizing a Denominator

You have developed some patterns to help determine the lengths of the sides of 30°- 60°- 90° and 45°- 45°- 90° triangles. This will enable you to solve similar problems in the future without a calculator. However, sometimes using the patterns leads to some strange looking answers. For example, when determining the length of side a in the triangle at right, using the pattern yields an answer of !!.

It is difficult to estimate the value of a number with a radical in the denominator, and it is also difficult to combine it with or compare it to other numbers. For these reasons, it is sometimes helpful to rationalize the denominator so that no radical remains in the denominator. Study the example below. Example: Simplify !

!.

Problems

1. !!= !

! 2. !

!= ! !

! 3. !

!= ! !

!

4. !!= !!

! 5. !

!""= !

! 6. !

! != !

!

Homework Sec 6.1.4 # 6-42, 6-43, 6-44, 6-46, 6-47