Unit NOTES Honors Common Core Math Da : Properties of Eponents Warm-Up: Before we begin toda s lesson, how much do ou remember about eponents? Use epanded form to write the rules for the eponents. OBJECTIVE Multipling Eponential Epressions 0 SUMMARY: m n a a OBJECTIVE Dividing Eponential Epressions (Remember: ) 6 0 SUMMARY: a a m n OBJECTIVE Negative Eponential Epressions: Simplif WAYS using epanded form AND the rule from OBJECTIVE 6 0 SUMMARY: n a OBJECTIVE Eponential Epressions Raised to a Power 6 ( ) ( ) ( m ) m n SUMMARY: ( a ) n SUMMARY: ( ab ) The lesson I. Find the value of in each of the following epressions. 7 / 6 ( ) ( ) / ( ) 6 / 6 ( )
Unit NOTES Honors Common Core Math II. Find the values of and in each of the following epressions. 6 6 ( 6) 6 ( 6 ) 6 ( ) 8 III. These will have more than correct solution pair for &. Find at least solution pairs. ( )( ) Possible Solutions Option : Option : Option : ( ) 8 Possible Solutions Option : Option : Option : Possible Solutions Option : Option : Option : IV. When there are so man rules to keep track of, it s ver eas to make careless mistakes. To help ou guard against that, it helps to become a critical thinker. Take a look at the epanded and simplified eamples below. One of them has been simplified correctl and there s an error in the other two. Identif the correctl simplified eample with a. For the incorrectl simplified eamples, write the correct answer and provide suggestions so that the same mistake is not made again. ( )( ) 0c d cd c d V. You ve seen some of the more common mistakes that can happen when simplifing eponential epressions, and ou ma have made similar mistakes in the past. For each of the net rows of problems, complete one of the problems correctl and two of the problems incorrectl. For the incorrect problems, tr to use errors that ou think might go unnoticed if someone wasn t paing close attention. When ou finish, ou ll switch papers with two different neighbors (one for each row) so that the can check our work, find, fi, and write suggestions for how those mistakes can be avoided. ( ) 6 ( ) ( ) 8 ( )
Unit NOTES Honors Common Core Math Da : Simplifing Radicals and Basic Operations Warm-up: Match the radical on the left with a radical on the right with the equivalent value. Use our calculator if necessar. The lesson: We are familiar with taking square roots ( ) or with taking cubed roots ( ), but ou ma not be as familiar with the elements of a radical. n r An inde in a radical tells ou how man times ou have to multipl the root times itself to get the radicand. For eample, in the equation 8 9, 8 is the radicand, 9 is the root, and the inde is because ou have to multipl the root b itself twice to get the radicand(9 9 9 8). When a radical is written without an inde, there is an understood inde of. 6? Radicand = Inde= Root is because 6? Radicand= Inde= Root is because You can use our calculator to do this, but for some of the more simple problems, ou should be able to figure them out in our head. Reminder: To use our calculator: Step : Tpe in the inde. Step : Press MATH Step : Choose : Step : Tpe in the radicand. You Tr: 8 96m n 8 v
Unit NOTES Honors Common Core Math BUT not ever problem will work out that nicel! Tr using our calculator to find an eact answer for = The calculator will give us an estimation, but we can t write down an irrational number like this eactl it can t be written as a fraction and the decimal never repeats or terminates. The best we can do for an eact answer is use simplest radical form. Here are some eamples of how to write these in simplest radical form. See if ou can come up with a method for doing this. Compare our method with our neighbor s and be prepared to share it with the class. (Hint: do ou remember how to make a factor tree?) 8 Simplifing Radicals: ) ) ) Eamples: 6 8 8 80n 96 8 86 0 8 6 6 8 z 9 7 z 87 z
Unit NOTES Honors Common Core Math Multipling Radicals: When written in radical form, it s onl possible to write two multiplied radicals as one if the inde is the same. As long as this requirement is met, ) multipl the ) multipl the ) Simplif! 8 0 6 8 0 8 0 7 9 0 8 9 Adding and Subtracting Radicals: You ve been combining like terms in algebraic epressions for a long time! Show our skills b simplifing the following epressions. = 6 = Usuall we sa that like terms are those that contain the same variable epression, but the can also contain the same radical epression. When ou add or subtract radicals, ou can onl do so if the contain the same inde and radicand. Just like we don t change the variable epression when we add or subtract, we re not going to change the radical epression either. All we are going to do is add or subtract the coefficients.
Unit NOTES Honors Common Core Math 6 Alwas simplif the radical before ou decide that ou can t add or subtract. + + + - 7-0 08 6 a 8a 9 0a 7 a 6 7 6 8 0 6 8
Unit NOTES Honors Common Core Math 7 Da : Rational Eponents & Radicals Warm-up (Properties of Eponents with FRACTIONS!) Even though the seem more complicated, fractions are numbers too. You can use all the same properties with fraction (rational) eponents as ou can with integer eponents. Write down those properties first. Work with a partner to write each epression in simplified form (one coefficient, each base used onl once with one eponent each, no negative eponents). Each of ou should complete one column, taking turns. Check each other s work.
Unit NOTES Honors Common Core Math 8 Raising a number to the power of ½ is the same as performing a familiar operation. Let s take a look at the graph of to discover that operation. Step : Tpe into the = screen on our graphing calculator. Step : Look at the table of values generated b this function. Verif that ou have the same values as the rest of our class. (It is ver eas to make a mistake when ou tpe in the eponents here!) Step : Discuss with our classmates what ou believe to be the relationship between the and values in the table. Where have ou seen this relationship before? Summarize our findings in a sentence. Step : Tpe into the = screen on our graphing calculator. Step : Look at the table of values generated b this function. Verif that ou have the same values as the rest of our class. (It is ver eas to make a mistake when ou tpe in the eponents here!) Step 6: Discuss with our classmates what ou believe to be the relationship between the and values in the table. Have ou seen this relationship before? Summarize our findings in a sentence. Step 7: Tpe into the = screen on our graphing calculator. Step 8: Adjust our table so that the values go up b ½ and begin at 0. Verif that our table contains the same values as the rest of our class. Step 9: Discuss with our classmates the pattern ou see. Use the table below to help ou see the pattern. (One row has been completed for ou). Summarize our findings in the space beside the table. X (eponent) X (eponent) as a fraction with a denominator of Y ( ) Rewrite Y as a power of with fraction eponents Rewrite Y as a power of 0.. / ( ).. How could ou use this pattern to find the value of / 6? Check our answer in the calculator. How could ou use this pattern to find the value of / 7? Check our answer in the calculator. How could ou use this pattern to find the value of / 8? Check our answer in the calculator.
Unit NOTES Honors Common Core Math 9 Step 0: Generall speaking, how can ou find the value of an epression containing a rational eponent. Use the epression m/ n a to help ou in our eplanation. You tr: Rewrite each of the following epressions in radical form. ( 7) ( 6 ) -9/8 a 7 ( ). Now, reverse the rule ou developed to change radical epressions into rational epressions. ( 6) 7 ( ) 7 ( 9 ) 7 ( ) Earlier in this unit, ou learned that when written in radical form, it s onl possible to write two multiplied radicals as one if the inde is the same. However, if ou convert the radical epressions into epressions with rational eponents, ou CAN multipl or divide them (as ou saw in our warm-up)! Give it a tr Write our final answer as a simplified radical.
Unit NOTES Honors Common Core Math 0 How does the idea of simplifing radicals relate to the idea of rational eponents? There are several was to approach this. Develop our own method for calculating simplest radical form of an epression without converting to radical form until the ver last step! Describe our method for simplifing radicals from rational eponents. Share our method with the class.
Unit NOTES Honors Common Core Math Da : Modeling with Eponential Functions: Solving Equations with Rational Eponents and Radicals Warm-up: You know a lot about inverses in mathematics we use them ever time we solve equations. Write down the inverse operation for each of the following (there could be more than one correct answer) and then give a definition for inverse in our own words. If ou get stuck, it ma be helpful for ou to write the epression out and think what ou would do to solve an equation that had that epression on one side of the equation. The phrase Is the epression And its inverse is adding to a number Subtracting from a number subtracting 7 from a number multipling a number b ½ Multipling a number b dividing a number b squaring a number Taking the square root of a number Raising a number to the th power Taking the th root of a number Raising a number to the power An inverse is Da : Modeling with Eponential Functions: Solving Equations with Rational Eponents and Radicals
Unit NOTES Honors Common Core Math The lesson: Yesterda s problems onl had one step. You cannot do this step until the radical or rational eponent is isolated on one side of the equation. You can isolate the radical using the inverses discussed at the beginning of the lesson. There are also some problems below in which the rational eponent or radical is applied to the entire side of the equation. Onl in these situations will ou undo the rational eponents or radicals first. Before solving the entire problem, make sure ou know what the first step will be.
Unit NOTES Honors Common Core Math Da : Modeling with Eponential Functions: Solving Equations with Rational Eponents and Radicals Warm-up: Solve for the missing variable.... Part Solving equations with radicals: The net few problems are different. We re going to come across some equations that have no solution and some that have two solutions. Remember, ou can alwas check our answers b substituting our solution into the equation to make sure it works. In fact, ou reall need to check our answers to these problems! When we solve an equation correctl, but the answer doesn t work when we check it, we call the solution etraneous. Applications of Equations with Rational Eponents or Radicals.. The distance between two points is. If one of the points is located at (, ) and the other point has a -value of -, what are the possible -values of the other point?
Unit NOTES Honors Common Core Math. The volume of a sphere is. If the formula what is the radius of the sphere? V ( /) r is used to calculate the volume of a sphere,. The equation v.r allows ou to calculate the maimum velocit, v, that a car can safel travel around a curve with a radius of feet. This is used b the Department of Transportation to determine the best speed limit for a given stretch of road. If a road has a speed limit of mph, what is the tightest turn on that road? Da 6: Quiz Da Warm-up: Simplif the following: Space for Practice Problems:. 9 6. 6 8 0. 6 Solve the following:. 8. ( ) 7 6. a a