INSTRUCTOR GUIDANCE EXAMPLE: Week Three Discussion Simplifying Radicals 1. Simplify each expression using the rules of exponents and explain the steps you are taking. 2. Next, write each expression in the equivalent radical form and demonstrate how it can be simplified in that form, if possible. 3. Which form do you think works better for the simplification process and why? #51. (2-4)1/2 2 The exponent working on an exponent calls for the Power Rule. The exponents multiply each other. -4*1/2 = -2 so the new exponent is -2. (-4*1/2) 2-2 1 22 The negative exponent makes a reciprocal of base number and exponent. The final simplified answer is . This is the principal root of the square root of 2-4. 1 4 1 81x 12 4 #63. 20 y 4 14 12 14 3 x 1 y 20 4 3x 3 y5 The Power Rule will be used again with the outside exponent multiplying both the inner exponents. 81 = 34 4*1/4 = 1, 12*1/4 = 3, and 20*1/4 = 5 All inner exponents were multiples of 4 so no rational exponents are left. 2 8 3 #89. 27 First rewrite each number as a prime to a power. 2 23 3 3 3 Use the Power Rule to multiply the inner exponents. The negative has to be dealt with somewhere so I will put it with the 2 in the numerator. 23 3 2 3 3 2 3 3*2/3 = 2 in both numerator and denominator. 22 3 2 4 9 The squaring eliminates the negative for the answer. It turns out that the examples I chose to work out here didn't use all of the vocabulary words and required one which wasn't on the list. Students should be sure to use words appropriate to the examples they work on. 4 3 (8) 3 3 2( 12 x 2 x) For this assignment I will need you to simplify and compare equivalent expressions written both in radical form and with rational (fractional) exponents, there are two problems that need to be completed and FIVE vocabulary words that need to be utilized throughout the post in describing solution and/or outcome; Principal Root Product Rule Quotient Rule Reciprocal nth Root Make sure to simplify each expression using the rules of exponents and examine the steps you are taking. This must be 250 words at least in length, and please make sure description/incorporation of words is original (plagiarism pings). My teacher grades on detail to ensure there is an understanding of the math, so that's important I have attached an example of what is expected in this week's post. Thank you so much and just let me know if you have any questions :D 1) 8 4 3 2 1 8 1 1 3 4/3 2 4 3 3 1 2 4 4 , using definition of Reciprocal to rewrite the base with positive exponent , as cube root of 8 is 2 so use the definition of nth root to rewrite 8 as 2 24 3 , as we have exponent working on exponent so use Power rule and multiply the exponents , exponent will be simplified as 1 1 4/3 4 3 4 3 , use Product rule to seperate the negative sign 1 4 , as forth root of 16 is 2 so use the definition of nth root to rewrite 2 as 16 and the even power 16 of 1 is always 1. 2) 3 2 3 12 x 3 2 x 3 2 3 12 x 3 2 3 2 x , use Distributive property to remove bracket 3 2 12 x 3 2 2 x , use Product rule to combine terms with like exponent by multiplying the base 3 24 x 3 4 x , perform the multiplication and simplify the expression inside radicals 3 233 x 3 4 x , factor 24 as 24 23 3 3 23 3 3 x 3 4 x , use Product rule to separate 23 term from 3 x term inside the radical 2 3 3 x 3 4 x , use definition of Principal cube-root and rewrite 3 23 as 2 1) 8 4 3 1 8 1 2 3 4/3 1 2 4 3 3 1 2 4 4 , using definition of Reciprocal to rewrite the base with positive exponent , as cube root of 8 is 2 so use the definition of nth root to rewrite 8 as 2 24 3 , as we have exponent working on exponent so use Power rule and multiply the exponents , exponent will be simplified as 1 1 4/3 4 3 4 3 , use Product rule to seperate the negative sign 1 4 , as forth root of 16 is 2 so use the definition of nth root to rewrite 2 as 16 and the even power 16 of 1 is always 1. 2) 3 2 3 12 x 3 2 x 3 2 3 12 x 3 2 3 2 x , use Distributive property to remove bracket 3 2 12 x 3 2 2 x , use Product rule to combine terms with like exponent by multiplying the base 3 24 x 3 4 x , perform the multiplication and simplify the expression inside radicals 3 233 x 3 4 x , factor 24 as 24 23 3 3 23 3 3 x 3 4 x , use Product rule to separate 23 term from 3 x term inside the radical 2 3 3 x 3 4 x , use definition of Principal cube-root and rewrite 3 23 as 2 1) 8 4 3 1 8 1 2 3 4/3 1 2 4 3 3 1 2 4 4 , using definition of Reciprocal to rewrite the base with positive exponent , as cube root of 8 is 2 so use the definition of nth root to rewrite 8 as 2 24 3 , as we have exponent working on exponent so use Power rule and multiply the exponents , exponent will be simplified as 1 1 4/3 4 3 4 3 , use Product rule to seperate the negative sign 1 4 , as forth root of 16 is 2 so use the definition of nth root to rewrite 2 as 16 and the even power 16 of 1 is always 1. 2) 3 2 3 12 x 3 2 x 3 2 3 12 x 3 2 3 2 x , use Distributive property to remove bracket 3 2 12 x 3 2 2 x , use Product rule to combine terms with like exponent by multiplying the base 3 24 x 3 4 x , perform the multiplication and simplify the expression inside radicals 3 233 x 3 4 x , factor 24 as 24 23 3 3 23 3 3 x 3 4 x , use Product rule to separate 23 term from 3 x term inside the radical 2 3 3 x 3 4 x , use definition of Principal cube-root and rewrite 3 23 as 2