6-36. FRACTIONAL EXPONENTS What happens when an exponent is a fraction 6-36. FRACTIONAL EXPONENTS What happens when an exponent is a fraction? Consider this as you answer the questions below. 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  in the exponent appear to have? 3, 7, 10 -> square root Based on your observation in part (a), predict the values of 41/2 and (71/2)2.  Then confirm your predictions using your calculator. 2, 7 Warm Up

6.1.4 Radicals and Fractional Exponents HW: 6-42,43,44,47 6.1.4 Radicals and Fractional Exponents December 6, 2018

Objectives CO: SWBAT use fractional exponents to simplify. LO: SWBAT rewrite radical expressions with rational (fractional) exponents and vice versa.

6-37. Danielle wants to understand why 91/2 is the same as 9 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. Using Danielle’s logic, what is the value of 161/2? Confirm your answer using your calculator. 4∙4 1/2 -> 4 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. 2∙2∙2 1/3 -> 2 5∙5∙5 1/3 -> 5 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.  Share any insight with your team members. 3∙3∙3 2/3 -> 9 2∙2∙2∙2∙2 3/5 -> 8 5∙5 3/2 -> 125 Another name for x1/3 is “cube root”.  This can be written  3 𝑥 .  What would be the notation for x1/5?  What should it be called? 5 𝑥 -> Fifth root Together/Team

Together ( 3 )4 (31/2)4 = 34/2 = 32 = 9 97/2 ( 9 )7 = 37 = 2,187 3 2 5 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 different ways and if possible simplify.  For example,  2 5 = (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. ( 3 )4 (31/2)4 = 34/2 = 32 = 9 97/2 ( 9 )7 = 37 = 2,187 3 2 5 (21/3)5 = 25/3 642/3 ( 3 64 )2 = 42 = 16 Together

Rewrite in other form (rational exponent or radical), then simplify. 16 3/4 4 16 3 = 2 3 =8 5 6 5 6 1/2 = 5 3 =125 9 3/2 9 3 = 3 3 =27 3 2 12 2 1/3 12 = 2 4 =16 Hot potato

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 = 3 1 2 .  Addison says, “Well, let’s assume there is some exponent that will give us the square root.” Addison writes the equation   3 = 3 𝑥 .  Imani says, “I think we should start by getting rid of the square root.”  What operation will “undo” the square root? Squaring Starting with Imani’s idea, solve the equation  3 = 3 𝑥  for x.  What does your answer mean? 3 = 3 𝑥 3= 3 𝑥 2 3 1 = 3 2𝑥 1=2𝑥 1 2 =𝑥 Can you extend the argument to other radical expressions?  Use a similar technique to demonstrate that  3 17 = 171/3. 3 17 = 17 𝑥 17= 17 𝑥 3 17 1 = 17 3𝑥 1=3𝑥 1 3 =𝑥 1&4:c, 2&3:b