Unit 2: Exponents and Radicals

Transcription

1 Unit 2: Exponents and Radicals Student Tracking Sheet Math 10 Common Name: Block: What I can do for this unit: After Practice After Review How I Did 2-1 I can explain the relationship between exponents and radicals, including indexes, for positive, negative, and zero exponents. 2-2 I can use exponent laws (product, quotient, and power of power) for integers. 2-3 I can apply exponent laws with fractional and negative exponents. Code Value Description N Not Yet Meeting Expectations I just don t get it. MM Minimally Meeting Expectations Barely got it, I need some prompting to help solve the question. M Meeting Expectations Got it, I understand the concept without help or prompting. E Exceeding Expectations Wow, nailed it! I can use this concept to solve problems I may have not seen in practice. I also get little details that may not be directly related to this target correct.

2 Unit 2: Exponents and Radicals Day 1 Math 10 Common 2-1 I can explain the relationship between exponents and radicals, including indexes, for positive, negative, and zero exponents. Radicals are the name given to finding a root of any degree. Last unit we looked at square and cube roots, but there are many other types of roots as well. We write radicals in the form: where is the index which tells us what root we are looking for, and is the radicand, which is the number being rooted. In the last unit we looked at radicals with index 2 (square roots) and 3 (cube roots). Note that: = and Thus = 32=2 because 2 =32 =, etc. Review: Evaluate the following radicals. (No calculator) 1) 9 2) ) 125 4) 16 Rewrite the following radicals in exponential form. ie. 25=5 is rewritten as 5 =25. (No calculator) 5) 121 6) ) 8) Estimate each radical (ie. 12 is between 3 and 4). (No calculator) 9) 8 10) 20 11) ) 100 Place each of the following radicals on a number line using your calculator. 13) 14) 25 15) 0.53

3 16) ). 18) 0.95 Exponents are operations performed on a base. If we examine!, is known as the base and is the exponent. Together they are known as a power. The exponent () tells us how many times to multiply the base (), by itself. For example, 5 =5 5 5=125. Observe the pattern for powers of 2: 2 =16,2 =8,2 =4,2 =1. We notice that as we go down by each power, we re dividing the previous number by the base (2). : 2 % =1,2 & =,2& = (or '). Complete the following patterns. (No calculator) 19) 3 =81 3 =27 3 = 3 = 3 % = 3 & = 3 & = 3 & = 20) = 4 % = 4 & = 4 & = 4 & = 21) 5 =625 5 = 5 = 5 = 5 % = 5 & = 5 & = 5 & From the above patterns, we see that % = =1. In other words, anything raised to the zero power is 1. Note that 0 % is the only exception to this (it is not defined because we can t divide by 0). We also see that &! = for every base except 0, again because we can t divide by 0. When we ) simplify powers, we write them with only positive exponents. Evaluate the following. (No calculator) 22) ( 18) % 23) ) (341) % 25) 8 % 26) * + % 27) *, +

4 28) ( 3) % 29) * +% 30) * + 31) Which of the following is equivalent to 9? ( 3), 3,( 3) &, 3 & 32) Which of the following is equivalent to? ( 4), 4,( 4) &, 4 & 33) Which of the following is equivalent to? ( 2), 2,( 2) &, 2 & 34) ( 5) 35) 5 36) 3 37) 38),' 39) ( 6) 40) 9 41) 51 42) ( 4) Place the following radicals on a number line without using your calculator. 43) 2 5, 3 2, 3 2,2 6, and 21 44) 4 2, 3 3, 4 2,3 3, and 5

5 Unit 2: Exponents and Radicals Day 2 Math 10 Common 2-2 I can use exponent laws (product, quotient, and power of power) for natural numbers. Review: Evaluate the following radicals without a calculator. 1) 169 2) 64 3) 51 4) 243 ) 5) 51) 6) 2) When we multiply powers with the same base, we add the exponents. For example: 3 3 =3 3) )=3 =3 Exponent Product Law: = Simplify and write without brackets or negative exponents. 7) 5 5 8) 9) ) ) 12) 3! 3 "! When we divide powers with the same base, we subtract the exponents. For example: 2 2 = =2 = Exponent Division Law: = Simplify and write without brackets or negative exponents. 13) ) $% "& $! 15) '! 16)! 17) ) () )! ) 19) *+ +,- 20),'! 21) & " We can also take the power of a power. ) = ) ) )= ) ) )= = Exponent Power of a Power Law: ) = Simplify and write without brackets or negative exponents.

6 22) 5 ) 23) 3 ) 24) /$! 0 & $ ' 25) 4 ) $ 26) 3 ) 3 27) / 0 1 * If we take the exponent of a product: 35 ) =3 5 =3 25 =75 (Note that the exponent doesn t apply to the 3 because it is outside the brackets.) Exponent Power of a Product Law: 2) = 2 Simplify and write without brackets or negative exponents. 28) 52 ) 29) 33 4 ) 30) (/$! 0 & ($) 31) 0.8 ) 32) 6 $'& $ 7 33) / ! )! Applying our laws to fractions: : 2 3 ; = 2 3 Exponent Power of a Quotient Law: 6 7 = < ) ) < Simplify and write without brackets or negative exponents. 34) ) ) )! )! 37) 3 = ) 38) ) / )! 0! / '" )0! )! ) '& 40) / '! 0! / '" 0 41) /$'! 8 '" 0 /$ 1 8 ' 0! 42) 6!! ) '"! '! ) '" $8 '& ) '! ) 7 6 '" ' )! 7 )!

7 Unit 2: Exponents and Radicals Day 3 Math 10 Common 2-3 I can use exponent laws (product, quotient, and power of power) for natural numbers. Review: Evaluate the following without a calculator. 1) 2 5 2) 3) ) 4) 5) 35 ) 6) From earlier, we saw that =! ". Thus 5# = (the exponent only applies to the # and not to the 5). Simplify and write without brackets or negative exponents. 7) 2 8) # & 9) 6 10) 3 ) 11) 4 ) 12) * 13) 5 )! 14) 7) 15) 5) 16) 2# & ) 17) 3 - )! 18) 3 & -) &

8 We know that both. / 0 Thus 27 = 27 =3 =9. And 16 =! =!!3!3 = and =! =! *. Evaluate each of the following without a calculator. =. Thus / =. In general, "= /. 19) 169 / 20) ) 81 " 22) 4 / 23) 4 / 24) 9 / 25) 8 26) 32 27) 81 / 4 28) ) 125 / 30) 27 Simplify and write in exponential form without brackets or negative exponents. 31) 3 / 32) #!5 4 33) 2 & 34) 3 / 3 35) 7 / 7 36) # #

9 37) ) * 8 * / * 8 39) & 40) / 41). * 0! 42).!3 *! 0 43).!3 0 / 44). 3 &) 0 45). * 0 Identify the errors and correct the answer. 46) 3 +3 =3 47) 2 3 ) 2 )=2! 48) 5 5 =5 49) 3 = 9 50) 51) ) =!5 =1

10 Solve the following problems: 52) Here is an expression for the percent of caffeine that remain in our body : hours after you drink a caffeine beverage: ) a) Use the expression above to determine the percent of caffeine that remains after 1.5 h. b) After how many hours does 50% of the caffeine remain? 53) In the late 1500s, Johannes Kepler developed a formula to calculate the time it takes each planet to orbit the sun (called the period). The formula is S 0.2U where S is the period in Earth days andu is the mean distance to the sun in millions of kilometres. The mean distance of Earth from the sun is 149 million kilometres. The mean distance of Mars from the sun is about 228 million kilometres. Which planet has the longer period Earth or Mars? Justify your answer. 54) The intensity of light at its source is 100%. The intensity, V, at a distance W centimetres from the source is given by the formula V=100W. Use the formula to determine the intensity of light 23cm from the source. 55) There is a gravitational force, X newtons, between the Earth and the moon. This force is given by the formula X= !! )YZ[, where Y is the mass of the Earth in kilograms, Z is the mass of the moon in kilograms, and [ is the distance between the Earth and the moon in metres. The mass of the Earth is approximately & kg. The mass of the moon in is approximately ^22 kg. The mean distance between them is approximately km. What is the gravitational force between the Earth and the moon?

11 Unit 2: Exponents and Radicals Day 4 Math 10 Common Review Only use a calculator when instructed. Evaluate the following radicals. 1) 225 2) ) 27 4) 625 Estimate each radical. (ie. 12 is between 3 and 4). 5) 51 6) 125 7) ) 20 Place each of the following radicals on a number line using your calculator. 9) 10) ) 0.71 Complete the following patterns. 12) ) 8 = 8 8 = 8 = 8 = 8 = 8 = Evaluate the following. 14) ) ) 125y) 8

12 17) # $ % 18) # % 19) 5 20) 3) 21) 5 22) Simplify and write without brackets or negative exponents. 23) & & 24) 7 7 ( 25) ) * ) 26) & & 27) 4) 3) 28) ),-.,- 30) 31) ) & ) 33) 6) ) 34) #.. % 35) 5 36) 5) / ) 37) # % Evaluate each of the following. 38) 81 39) 25 40) 16

13 41) 81 42) 27 43) 64 44) ) 4 46) 125 Simplify and write in exponential form without brackets or negative exponents. 47) 5 48) 5-49) & 50) ) ) ) # $ % 54) # $ % 55),, 4, 4- Solve the following: 56) The circulation time is the average time it takes for all the blood in the body to circulate once and return to the heart. The circulation time for a mammal can be estimated by the formula I where I is the circulation time in seconds and 1 is the body mass in kilograms. Estimate the circulation time for a mammal with mass 85 kg. 57) Suppose you want $5000 in 3 years. The interest rate for a savings account is 2.9% compounded annually. The money, R dollars, you must invest now is given by the formula R = ). How much must you invest now to save $5000 in 3 years?

14 Unit 2: Exponents and Radicals Key Math 10 Common Day 1: ,3,1, 1 3,1 9, ,16,4,1,1 4, 1 16, ,25,5,1,1 5, 1 25, Day 2: ! 26 3 " 27 2 " " # 30 $ " & # 41 ' & Day 3: & " 1 10 " 2! 3 7# ' ( )%* % ' % % 16 +* , $ +- +%, % 32 % ( ' " $ 34 3 $ ( ( %* / 39 (+ +% 40 5 %0 +$ " ) a)81% b) 5 hours ' 53) Earth 364 days, Mars 688 days 54) 0.19% 55) 2110! 43 Day 4: 45 " ,16,4,1, 1 4, 1 16, , $ -, 8, 8, 8, " % " 29! 30 5 ' &' ( ( 50 3 % * ) sec 57) $4589