Everything I know about exponents

Prescribed Learning Outcomes for Exponents:

    1. Represent repeated multiplication with exponents:  The first step to represent a repeated multiplication with powers is to count how many time the number is multiply by it self. My example for this step is 17 x 17 x 17 x 17, you can count how many times 17 is times by it self, the answer is 4 times. The next step is now representing this with exponents, the way that I do this is taking the number that was timed by it self as the base and taking the number of how many times it was timed as the exponent. The example for this is 17 is the base and the 4 is the exponent = 17^4
    2. Describe how powers represent repeated multiplication: How I describe the way powers can be represent as repeated multiplication is very similar to learning outcome number 1. You start by identifying the base and the exponent of the power. The example I have; 7^5, the base is 7 and the exponent is 5. With the information about the base and the exponent, you can start by taking the base and multiplying it by it’s self the amount of times the exponent said. My example is with your 7 and 5, the 7 being the base, you will times 7 x 7 x 7 x 7 x 7 because the exponent is 5.
    3. Demonstrate the difference between the exponent and the base by building models of a given power:  For this learning outcome, I have two image to show how it works.         On the left is cube and let’s said that it is = to 7^3 and the square is equaled to 7^2. The number 7 is representing the side length of the cube and a side length of the square. The difference is my two examples is 7^3 is that the answer is represent the volume of a cube. The 7^2 is represent the area of a square.
    4. Demonstrate the difference between two given powers in which the exponent and the base are interchanged by using repeated multiplication: Where the exponent and base are interchanged will not equal the same answer. The reason behind this is that you will always multiply the base by it’s self the amount of time of the exponent. So, with that 3^4 will not be the same as 4^3 because they have different repeated multiplication. 3^4 = 3 x 3 x 3 x 3 (the answer being 81), when 4^3 = 4 x 4 x 4 (the answer being 64). The answer is difference making the powers have a difference.
    5. Evaluate powers with integral bases (excluding base 0) and whole numbers exponents:  You can evaluate powers with integral base and whole numbers exponents the same way. My examples are -3^4 and 5^8. -3^4 =  -3 x -3 x -3 x -3 = 81 and 5^8 = 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 = 390625. The both of those powers use the same steps, step one find the power and the exponent in the power. Next take the power and multiply it by it’s self the amount of times of the exponent. Then after doing the multiplication, you have your answer.
    6. Explain the role of parentheses in powers by evaluating a given set of powers: The role of parentheses can change an answer to a power very drastically. My examples are (-2)^4 and -2^4, as you can see they look like almost the same power with one having parentheses and the other not having them. (-2)^4 in repeated multiplication is (-2) x (-2) x (-2) x (-2) = 16, when $latex -2^4 in repeated multiplication form is -1 x -2 x -2 x -2 x -2 = -16. The answers are completely opposite.
    7. Explain the exponent laws for multiplying and dividing powers with same base:  The exponent law for multiplying with the same base is to keep the base. Then add the exponents and multiply the coefficients. For example the equation; 2(x)^3 x 3(x)^7, you would first add 3 and 7 because they are the exponents, which equals 10. After that you are going to times the coefficients; 2 x 3 = 6. The answer will be 6(x)^(10). The exponent law for dividing powers of the same base is when you keep the base, subtract the exponents, and then divide the coefficients.  My example is 15(y)^4 divide by 3(y)^2, the first step would be to subtract the exponents; 4 – 2 = 2. Then you would divide the coefficients; 15 divide by 3 = 5. The answer to 15(y)^4 divide by 3(y)^2 = $latex 3(y)^.
    8. Explain the exponent laws for raising a product and quotient to an exponent: When you are raising a product, you are really doing a power to a power. For example (8^2)^4, the law is to keep the base the same and multiply the exponents. Your base is 8 so you keep it, and then multiply 2 x 4 which equals 8. So now you have 8^8 which equals 16 777 216.
    9. Explain the law for powers with an exponent of zero:  This law is pretty straight forward to explain, no matter what is your base if your exponent is 0 the answer will always be 1. My example for this law is 19^0 and the answer is going to 1.
    10. Use patterns to show that a power with an exponent of zero is equal to one: I am going to start with the base of 6 and an exponent of 9. 6^9 = 10077696, 6^8 = 1679616, 6^7 = 279936, 6^6 = 46656, 6^5 = 7776, 6^4 = 1296, 6^3 = 216, 6^2 = 36, 6^1 = 6, 6^0 = 1. The way that we can use the pattern to explain the zero law. If you divided the answer of 6^9 by 6 it will be the answer of the next 6^8. 10077696 divided by 6 equals 167916, if you keep doing that until 6^1  = 6, six divided by 6 equals 1. That is the reason any base to the exponent of 0 equals 1.
    11. Explain the law for powers with negative exponents:  The law for is to find the reciprocal of the base of the power, then keep the exponent with the original base, and next drop the negative sign. My example is 4^-3. The way to find the reciprocal is first to put the power in a fraction; 4^-3/1. Then the reciprocal will be 1/4^3, however now you drop the negative sign. The answer to my example is 1/64.
    12. Use patterns to explain the negative exponent law: For showing how to explain the negative exponent law I am going to start of with the power 3^3, we know this equals 27. The next one would be 3^2 which equals 9. After 3^1 equals 3, and then 3^0 equals 1. With the start of the patterns we can already figure out how to do the next bit of the pattern. You can take the 27 and divide it by 3, which equals 9, the next answer of the pattern. If you continue doing that, 9 divide by 3 equals 3, 3 divide by 3 equals 1. So now, if you take 1 and divide it by 3 it equals 1/3, which is also the answer to 3^-1. Next 1/3 divide by 3 equals 1/9, the answer of 3^-2.
    13. I can apply the exponent laws to powers with both integral and variable bases: First of all an integral base is a base that could be any number, such as …-2, -1, 0, 1, 2, 3… A variable base is a base that is represented by a letter because the value of the base is unknown. My example for the integral base is (-4)^2 x (-4)^3. The first step is to use the product law, add the exponents, 2 + 3 = 5, keep the base. Now the power is (-4)^5, which equals -1024. The example I have for variable bases is y^-2 divide by y^-4. First keep the base, then subtract the exponents; -2 – (-4) = 2. Now the answer is y^2.
    14. I can identify the error in a simplification of an expression involving powers:  In this expression I can identify the error in simplification; 3^5 x 3^3 = 3^7. The way I can find the error is by following the product law. The first step is to keep the base; 3. Then you add the exponents; 5 + 3 = 8. The answer is 3^8.
    15. Use the order of operations on expressions with powers: When using the order of operations, follow the acronym; BEDMAS: brackets, exponents, division, multiplication, addition, subtraction. My example is 2^5 – 7 x  (-5 + 2)^3. The first thing you need to do is the adding in the brackets; -5 + 2 = -3. Next, you are going to do the exponents; (-3)^3 = -27, and 2^5 = 32. After that you are going to do the subtraction; 32 – (-27) = 59. The answer to 2^5 – 7 x (-5 + 2)^3 = 59.
    16. Determine the sum and difference of two power:  Let start with the sum of two powers first. My example is 2^4 + 3^5, the first step is to solve each power. 2^4 = 16 and 3^5 = 243. The step is to add the answers of the powers. 16 + 243 = 259. When finding the difference of two powers, you have very similar step to finding the sum. The example I have is 2^93^2. Your first step again is to find each answer to the powers. 2^9 = 512 and 3^2 = 9. Now, subtract the two answers. 512 – 9 = 203.
    17. Identify the error in applying the order of operations in an incorrect solution:  The example I have for the incorrect solution is (-2 + 4)^6 – 5 x 2^3, and first you did the brackets; (-2 + 4) = 2. Then the exponents; 2^6 =  64 and 2^3 = 8. Next you did the subtraction; 64 – 5 = 59. Then finally you did the multiplication; 59 x 8 = 472. Now I am going to find out when the error is in the operation. The first you need to do is to follow BEDMAS. I know by looking at the operation, I need to do brackets, then exponents, then multiplication, and finally subtraction. Already I can see the error, when I subtract was when I was supposed to multiply. If I was to do it right 5 x 8 = 40. Then 64 – 40 = 24. The correct answer to the order of operations is 24.
    18. Use powers to solve problems (measurement problems):  My example is; if you had a 7cm side length of a shaded squared with a square that isn’t shaded on the inside that has a side length of 3cm, much of the shaded part is left. This question written out is 7^2cm3^2cm, and that would equal 49cm – 9cm = 20cm^2.
    19. Use powers to solve problems (growth problems):  The example for this learning outcome is a factory triples the number of toys each hour. There is 25 toys so how many toys will the factory make after five hours? 1 hour = 25 x 3 = 75. 2 hours = 25 x 3 x 3 or 25 x 3^2 = 225. 3 hours = 25 x 3 x 3 x 3 or 25 x 3^3 = 675. 4 hours = 25 x 3 x 3 x 3 x 3 or 25 x 3^4 = 2025. 5 hours = 25 x 3 x 3 x 3 x 3 x 3 or $latex 25 x 3^5 = 6075.
    20. Applying the order of operations with powers involving negative exponents and variable bases:  My example is (-10)^-x x (y). The first step is to do change it into y(-10)^-x. Then put it into the reciprocal of the base = 1/y(-10)^x

Digital Footprint

A Digital Footprint

How might your digital footprint affect your future opportunities?

It could have a positive impact on future opportunities when you have an appropriate footprint

  • My example is getting the job because of what you posted; community service you did, a cool school project, or something that you are proud of. Your posts could also show your values and positive family connections. It could also highlight your communication skills with how you respond to people’s post or what you write in your posts.
  • Why it would affect your future? If you have a good footprint that shows what kind of person you are, this may allow you to receive new opportunities in life. An employer would rather hire someone who shows positive things about themselves. Than having someone who they know nothing about.  

It could have a negative impact from having an impropriate footprint

  • The example I chose is losing a scholarship because of rude or impropriate things you put on social media; racist comments, photos of you doing inappropriate things, or just a lot of photos of you. Why it would affect your future: I always hear a lot of stories about people who lose scholarships because of the things that they posted in the past. You might have worked super hard on getting a scholarship for a sport you play. However  you said something on the internet that was mean or racist in the past. That could determine if you get the scholarship or the other person of equal skill gets it.

Describe at least three strategies that you use to keep your digital footprint appropriate and safe!

Prior to posting anything I would ask myself, would I be proud to show it to anyone?

  • I believe that is important to think about what you’re going to post before posting it. You need to take the time before you respond to posts. You want to keep a safe footprint. For example: Would you want your grandma to see what you posted? If you post something that you wouldn’t want your parents to see or your teachers to see, it’s likely something that isn’t appropriate. You should also always consider the words that you use when responding to someone’s posts. You don’t want to get involved with negatively or saying mean things to someone. You can always disagree with someone politely without be rude. Choose your words carefully. 

Checking with an adult or a parent

  • If you are not yet an adult, I think you should talk to someone who has more experience in life about what you are going to post. They might have advice on what you are posting and may be able to let you know if they think it will have a negative or positive affect on your future. Even if you are an adult it would be a good idea to get someone else to check if what you are posting is good for your image. Most importantly, trust your gut. If you’re unsure, don’t post!  

Think is it important or will it contribute to the world

  • Is it really that important to post a photo of everything that you are doing? Does the whole world need to know that you ate spaghetti for dinner? If it is important for you to post things about your life so old friends or family members can see what you been up to, try to think if they rather see a nice photo of you enjoying an activity or spending time with someone you care about. Than the same photo at different angles. Posting a million selfies can portray you as a self-absorbed person. You need to have a balance of what kinds of photos you post. Social media is a good place to express your opinions but be sure to know your facts.  

What did I learn that I would pass on to other students? How would you go about telling them?

What did I learn?

  • It may not seem important at the time but your digital footprint will have an impact on your entire future. So, it’s necessary to think before you post and put your best image on the internet. Just think, will what I’m doing on the internet have a positive or negative impact on my life.

How would I go about telling other students?

  • I believe having students listen to other people’s experiences about social media horror stories would help people realize the consequences. Students would get a clear idea of what would not be a good thing to post and how certain things might affect your future. Also having posters around the school that advertise be safe on the internet and keep an appropriative digital footprint would be a good reminder.

By Mackenzie C

First photo by Emelie Lundqvist; found on creative commons

Second, third, and fifth photo by Me

Forth photo:  https://1.bp.blogspot.com/-mcwdDEnwIu4/WSNVTQFcmEI/AAAAAAAALBs/rhnxnOqMf6c6BSgsE4XtSPBT4vyWJoO0gCLcB/s1600/345___big-grin-rosy-cheeks-emoji.png