Everything I Know About Exponents

1. Represent repeated multiplication with exponents 

When an expression is written in repeated multipcation, it is much easier to take the amount of factors of the base is turned into the exponent. This doesn’t work with 0 though which has a different outcome which is explained in a different question.

Examples: 6x6x6x6=6^4 and 3x3x3=3^3.

 3. Demonstrate the difference between the exponent and the base by building models of a given power, such as 2^3 and 3^2. 

The difference between the base and the exponent is that the base is the original number, while the exponent which is either written in a superscript next to the base and is how many factors of the base in which will be multiplied together.

6^7, the base is 6, the exponent is 7. It can also be written as 6 to the exponent of 7. That would mean 6 x 6 x 6 x 6 x 6 x 6 x 6. The whole thing of 6^7/6 to the exponent of 7 is known as the power. Though, 7^6 is different 7 x 7 x 7 x 7 x 7 x 7.

5. Evaluate powers with integral bases (excluding base 0) and whole number exponents.  

When dealing with bases, you will notice bases that are positive and negative. First of all, positive bases go as showed before where you simply multiply the base as many times as the value of the exponent (ex. 6^2 = 6×6 because the exponent is 2 and you are multiplying two 6’s together. Though, when dealing with negative bases, the answer and whether it is positive or negative depends on if the exponent is even or odd. If you have a negative base and the exponent is even, the negatives will cancel each other out, but if the exponents are odd, then the answer will be negative. (ex. (-5)^4 will be 625 while (-4)^3 will be -64). There are some interesting situations where you will see a negative power with no brackets around the base(-6^3) and the outcome to that will always be negative because the negative sign in this expression would be a coefficient because the exponent is inside the brackets. So if you were to write -6^3 as a repeated multipication, it would be written as -1 x 6 x 6 x 6.

7. Explain the exponent laws for multiplying and dividing powers with the same base.  

When multiplying powers with the same base, you keep the base the same, add the exponents, and if there are coefficients, multiply them.

2^2 x 2^4 = 2^6 and 3 (3)^4 x 2 (3)^2 = 6 (3)^6

When dividing powers with the same base, you keep the base the same, subtract the exponents, and if there are coefficients, divide them.

4^7\div/ 4^5 = 4^2 and 6 (3)^3\div 3 (3)^4 = 2 (3)^-1 = \frac{2}{3}.

9. Explain the law for powers with an exponent of zero 

Anything to the exponent of zero is equal to 1. This is because following the pattern that bases are just multiplied by themselves, they can also be divided by their base to find the number therefore if 2^2=4 and 4 \div2 = 2 and 2^1 is 2, then 2^0 must be 1 because 2^1 = 2 and \frac{2}{2} = 1.

3^0 = 1 and 35783297628762967589237^0 = 1.

11. Explain the law for powers with negative exponents.   

When dealing with negative exponents, you have to find the answer as if the exponent was positive and reciprocal it. Don’t just put a one over it because if you are dealing with fractions then you

Image result for exponents

will end up with a complex fraction and an incorrect answer.

3^-2 = \frac{1}{9} and x^-y = \frac{1}{x^y} and 2/3^-6 = 2 x 3^6.

13. I can apply the exponent laws to powers with both integral and variable bases.  

Product Law: When multiplying powers of the same base, you keep the base and add the exponents. If there are coefficients, multiply them. For example, 5 (3^2) x 4 (3^3) = 20 (3^5) = 20 x 243 = 4860 and xy^a*by^c=xby^a+c.

Power Law: If there is an equation with one base and two exponents, then the expression is solved differently than regular multiplication. Instead of adding the exponents, you multiply them. For example, (5^4)^3 = 5^12, and (x^y)^z = x^yz.

Quotient Law: When dividing powers of the same base, you keep the base and subtract the exponents. If there are coefficients, divide them. For example, 6 2^3\div 2 2^4 = 3 2^-1 = 3\frac{1}{2}. and yx^c * bx^a=ybx^a+c.

Negative Exponent Law: You need to reciprocal the answer of the power if the exponent positive. For example, 3^-4= \frac{1}{3^4} and x^-y = \frac{1}{x^y}. Also, \frac{2}{3}^-4= \frac{ 3^4}{2} and \frac{n}{m}^-r = \frac{m^r}{n}.

Power of 0 Law: Anything to the exponent of 0 = 1. 3647344327560274363454687067^0=1 and x^0=1. Also, x cannot equal 0, and 0^0 does not equal 1.

15. Use the order of operations on expressions with powers. 

Remember that you must do brackets first, then exponents. But, if there are exponents inside the bracket, then do the exponents there.

(2 x 3^2 + 8) / 13

(2 x 9 + 8) / 13

(18 + 8) / 13

26 / 13

= 2

17. Identify the error in applying the order of operations in an incorrect solution. 

If you mistake yourself and multiply/divide before applying the power, you will end up with the wrong answer.

(incorrect)

2(4)^5

8^5

32768

(correct)

2(4)^5

2(1024)

2048

As you can see, there is a major difference between the two answers.

19. Use powers to solve problems (growth problems) 

When doing growth problems, powers are used when showing how much something grew. The amount that you started with would be the coefficient and the base would be how much it grows by per said growing period and the exponent is what changes when the amount of time changes.

A colony of bacteria doubles every hour. There are 50 to begin with. How much will be there after 1 hour? 3 hours? 8 hours? Keep in mind it does not have to be hours. It could be seconds, minutes or even days, months, years, decades, centuries, etc.

So far, you have a consistent amount and how much it goes up by per hour, which is the coefficient and base. It is written like this: 50 x 2^n. n=how many hours it has been. So, to find how much there will be in one hour, you would write 50 x 2^1. 2^1 = 2 and 50 x 2 = 100. To find how much there will be in 3 hours, you would write 50 x 2^3. 2^3=8 and 50 x 8= 400. Lastly, to find how much there will be in 8 hours, you would write 50 x 2^8. 2^8=256 and 50 x 256 = 12800.

 

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Digital Footprint

1) How might your digital footprint affect your future opportunities?  Give atleast two examples.

There are many things that your digital footprint could affect. One big thing it could affect, especially right now, is your ability to get into certain colleges/universities. The admission officers may go through your digital footprint and look for you online, and they may find something that concerns them and makes them reject your application. Or, another example would be trying to get a job. Again, this is something big that your digital portfolio could affect, especially now, because many people at this school are legible to apply to jobs and may want to do so the first chance they get. This could also overall just affect your life at school or in public, if people recognize you from an internet post that maybe wasn’t appropriate or was not safe people may judge you. Or, if you post something not appropriate, someone may take that and bother you about it resulting in bullying or harrassment. Lastly, if you post something innappropriate, a threat of some sort, something that is harassing someone else, or cyberbullying, you could get a record with the police which could affect your future because now you have a record with the police and when employers and admission officers notice that then you have a much higher chance of being rejected from college/jobs.

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2) Describe atleast three strategies that you can use to keep your digital footprint appropriate and safe.

The first strategy that I’d say is probably the easiest to maintain it to simply think before you post, and know what you’re putting out there and just realize how important that is. If you can just manage your identity by just knowing what you’re putting out there and having a thought about what you’re posting then it’s pretty easy to keep things safe and appropriate. Well, when it comes to keeping it appropriate, then just post appropriate things, not too hard. The second strategy I would say is helpful is to use privacy settings. Make your content and posts private so that only a select amount of people can see it and comment on it. This can also block out any dangerous people and/or stalkers or possibly people who may take your content and do something bad with it. My third strategy would to use your social media and internet appropriately. Don’t use social media to post things that may get you into trouble. Also, don’t use it to cyberbully other because that isn’t quite smart either. First of all, thats simply unkind and you shouldn’t take time out of your day to make other people’s lives miserable. Secondly, someone may get mad at you for it or maybe screenshot it and use it against you one day. So always be careful with what you post because things could even be manipulated into something you totally didn’t mean and it could be used negatively against you. My last strategy that I think is significant is generally just being smart with how you use the internet and how you form your digital footprint. Don’t post mean/stupid things, don’t let random creepy people friend you on social media, don’t start messaging people you don’t know. There are so many ways to prevent these dangerous things, you just have to know how what those ways are.

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

I learned something really interesting about our cellphones. If you go to Settings>Privacy>Location Services>System Services>Significant Locations, you will find everywhere you have been to for the last month or so and it will tell you how long you were there for. If I were to tell my friends it would most likely be brought up in conversation when either talking about our phones, or this course and maybe discussing an assignment and that would remind me of what I learned. Something else interesting that I learned about is datamining. That is where ad makers find information about you and send specific ads to your phone that they think you’d be most interested in because of who you, how old you are, your gender, etc. My mom gets specific ads all the time but it’ll be things she is discussing in person then she will get an ad for it within a few minutes! Creepy!

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Sources:

Photo 1: Pexels.com

https://www.pexels.com/photo/man-and-woman-shaking-hands-1249158/

Photo 2: Pexels.com

https://www.pexels.com/photo/blur-display-electronics-hand-174938/

Photo 3: Pexels.com

https://www.pexels.com/photo/three-women-and-one-man-standing-while-using-phones-1638414/

Photo 4: Pexels.com

https://www.pexels.com/photo/angry-bad-john-art-black-and-white-emotion-709732/

Photo 5: Pexels.com

https://www.pexels.com/photo/three-people-using-smartphones-1471752/

Photo 6: Pexels.com

https://www.pexels.com/photo/smartphone-outside-hiking-technology-35969/