As the name of my post already gives out a lot of details and hints to what this post will be about, this post will be everything I know about exponents. I will be doing the even questions 2, 4, 6 etc up to 20 and my partner Aryan will be doing the odd questions so 1, 3, 5 etc. (The link to his blogpost is http://myriverside.sd43.bc.ca/aryanh2019/2019/11/08/everything-i-know-about-exponents/comment-page-1/#comment-10) This information that I am explaining is everything that I have learned during my math class and I hope you guys learn something new from my post. Also, if there is any errors and things that I should work on, feel free to tell me any comments would be greatly appreciated. Thank you and let’s get into it.

2) Describe how powers represent repeated multiplication

Within a power, there are two parts that are in the power, which are the base, and the exponent. The base is what number is used when multiplied by itself a certain number of times, and the exponent determines how many times the base is multiplied by. So for example, If a power is 2 to the exponent of three, or 2³, 2 is the base and 3 is the exponent so in this case you would multiply 2 by itself three times like so: 2 x 2 x 2 which would equal 8. So as you can see, the power represents multiplication because the exponent within the power represents how many times you would multiply the base by. Another example would be 3 to the exponent of 4 which in this case 3 is the base and 4 is the exponent so it would look like so: 3⁴= 3 x 3 x 3 x 3 = 81 because you would multiply three four times.

4) Demonstrate the difference between two given powers in which the exponent and the base are interchanged by using repeated multiplication, such as 2³ and 3².

This is something extremely common that many people get mixed up and confused on. If you take this example 3² and 2³, the difference here are the bases and the exponents. For 3² , in form of a repeated multiplication it would equal 3 x 3 which is just 9. The difference here is that 2³ is different when you turn it into a repeated multiplication, it would actually be 2 x 2 x 2 and I will explain why. Like I mentioned earlier in question 2, 3 to the exponent of 2 is another way of saying 3 x 3, so you are multiplying 3 two times because 3 is the base and 2 is the exponent. 2 to the exponent of three is 2 x 2 x 2 because 2 is the base which represents what number you are multiplying it by, and 3 is the power which represents how many times you would multiply the base by. Another example would be 4² and 2⁴ where 4² is 4 x 4, and 2⁴ is 2 x 2 x 2 x 2 which is different.

6) Explain the role of parentheses in powers by evaluating a given set of powers such as (-2)⁴, (-2⁴), and -2⁴.

A lot of people tend to get the role of parentheses or brackets mixed up when they are doing their questions or evaluations, and the difference between these three are actually very crucial and they affect your answer heavily. Firstly, we will start with (-2)⁴. When there are brackets that surround the number and not the exponent, it means that whatever is in the brackets is the base, and whatever is on the outside in smaller letters is the exponent, so for this one it would be (-2) x (-2) x (-2) x (-2) which would equal 16. But be careful because if the exponent is an odd number the result will always be negative, but if the power is even then the answer will be positive. Next, there is (-2⁴). In this question, you are detaching the negative symbol from the base which is 2, and the exponent which is four, you leave the negative sign to the side. So once you get your answer which is positive 16, all you would have to do now is attach the negative sign to make it -16, the negative symbol here acts as a coefficient where the negative symbol will be -1 and you add that to the question. This is the same with -2⁴, you will be evaluating 2⁴ first and then attaching the negative sign afterwards. When the negative sign is not a part of the base and is outside of the brackets like -(2⁴), there would be a coefficient of –1 on the outside, so if we took that as an example, it would be (-1) x (2) x (2) x (2) x (2).

8) Explain the exponent laws for raising a product and quotient to an exponent.

Starting with raising a product to an exponent, you can use an example, something as simple as (abc)². Using the power law, you are “distributing” or applying the exponent on the outside to each and every number/letter that is inside of the parentheses. So if you take this example, (abc)² would equal to a²b²c² because you are applying 2 to each of the variables. A more complex example would be something like (-3x⁴y⁵)². So using the power law, the first thing that you would do like we talked about is to apply the power on the outside to what is on the inside, so simplified it would look like ((-3)²)(x⁸)(y¹⁰). Now, why does x⁴ or y⁵ turn into x⁸ and y¹⁰? Because the power law states that you are multiplying the exponent on the outside of the brackets to the exponent that is on the inside of the brackets. Now, for raising a quotient to an exponent if you had something like a fraction where it looks like (⅔)², it would simplify to 2²/3² as you would distribute 2 to the two numbers in the fraction, and the answer would be ⁴/₉. 

10) Use patterns to show that a power with an exponent of zero is equal to one.

Surprisingly, there is actually a pattern for this and it’s pretty simple. This pattern is that you can take for example 2³, 2², 2¹, and 2⁰. 2³ equals 8, 2² is 4, 2¹ is 2 and 2⁰ is 1 so you can sort of see a pattern here that you are dividing each answer with the base each time which will always give you one in the end.  You can also use another example with 3³, 3², 3¹ and 3⁰ it’s the same pattern, first it’s 27, then 9, then 3, then 1, you are always dividing the answer by the base to obtain the next answer. The only exception is that you cannot make the base 0. You can make the base positive and negative  but you cannot make it 0 because the answer will always be 0. There is also a relation in between the quotient/product rule and this pattern because as an example we can use 2³ ÷ 2³, it would be 2³⁻³ which would equal to 2⁰ which is 1. This has a relation to this because instead of doing 2³ ÷ 2³ you can just evaluate for the two powers first which would be 8/8 which is also the same as 1. 

12) Use patterns to explain the negative exponent law.

Things from here start to get a little more tricky for many people, even myself. If any base is to the exponent that is negative, there is something else that you have to do. This law or rule is that you must reciprocal the base so that you can solve the question. The reciprocal is finding the opposite of the number, so if you were to find the reciprocal of 4 it would be ¼. So in this case as an example if there is something written out like 4⁻³, you wouldn’t just evaluate 4 to the power of 3 and then add a negative sign to the answer, there is something else. First off you would have to put this under a fraction of 1 (so like I mentioned previously, the reciprocal of 4³ which would be 1/4³) so if you took 4⁻³ as an example, a simplified form would be 1/4³. The reason why the negative exponent is turned positive is because you put it under a fraction, you are finding the reciprocal which is the opposite. So now you would evaluate which would simply be 1/64. Using patterns, you could use bases of 2, something like this: 2³, 2², 2¹, 2⁰, 2⁻¹, 2⁻², 2⁻³. Here you can see that the pattern is that every time the exponent is decreasing by 1. The answers would be 8, 4, 2, 1, ½, ¼, and ⅛. You see here that there is another pattern. When the exponent is positive the outcome becomes a whole, and when the exponent is negative is becomes a fraction because for powers with an exponent of negative something, you have to find the reciprocal. You can also see another pattern, is that you’re basically finding the reciprocal of the positive answer into a fraction over one so for example, if 2³ is equal to 8, 2⁻³ would just be ⅛, you are putting the answer over one.

14) I can identify the error in a simplification of an expression involving powers.

Common mistakes can happen all the time for anyone. Here is an example of an expression that a lot of people get very confused on. (-4p²q⁴)⁴. Someone would easily get mixed up on this question because when they are applying the exponent on the outside to the variables and the numbers on the inside, they especially get mixed up on the number, where they would ignore the brackets and then after they apply use the power law and etc. They would get -256p⁸q¹⁶. THIS IS NOT CORRECT. It may not seem like it but if you’re doing a more complex equation then you must pay attention to the negative symbol. When the exponent is applied to what is in the brackets (always depending on whether the power is even or odd, even positive odd negative), you have to attach the negative symbol with the number so if I were to do  (-4p²q⁴)⁴, when you apply the exponent it would look like this: (-4)⁴ and not (-4⁴), you are not applying it to the inside but just what the number is itself, whether it’s positive or negative. If you do it this way (-4⁴) it will give you always a negative number. If you do (-4)⁴ and the exponent is even, then the result it positive, and so in the end the answer would be 256p⁸q¹⁶.

16) Determine the sum and difference of two powers.

There is a very simple solution that works for finding the sum and the difference of two powers, and that is to just solve the powers first and then subtract and add them. Pretty easy right? As an example, we can take 2⁵ + 3² as an example for the sum. What you would do is find the answer for those two powers and then add them up, so the answer a little more simplified would be (2 x 2 x 2 x 2 x 2) + (3 x 3) which is 32 + 9 which is 41. For the difference (subtraction), we can use an example like 2⁶ – 3³. First you would solve, and then you would subtract the two answers. So step by step firstly it would be (2 x 2 x 2 x 2 x 2 x 2) – (3 x 3 x 3) which is then 64 – 27 which the final answer is 27. So as you can see the process is simple logic, there isn’t nothing too complex to it, it’s just finding the powers, simplifying like I did above and solving it whether it is addition or subtraction.

18) Use powers to solve problems (measurement problems)

For this question I will use shapes as an example. If there was a square and one of the sides was 3cm, and the question said to find the area these are the steps that you would take. The first step would be to realize that a square has equal sides, and then the next step would just be to multiply 3 x 3 to find the area of a square, which would just be 9cm² or just 3²cm² if you were to put it in terms of powers. Another example would be a right triangle which is a bit more complicated because you would use a formula called Pythagorean theorem to find the side of a right triangle (NOTE: the Pythagorean theorem does not apply to triangles that do not have a right angle). So, if there was a right triangle with two sides that you already knew, 3cm and 4cm and you had to find the longest side (the hypotenuse), what you would do is use this formula which is a² + b² = c². Now, we already know that a and b are 3 and 4, and so it would be 3² + 4² = c². Now it would be 9 + 16 = c². Then, you would add 9 and 16 which now gives you 25 = c². The final step would be to square root c so that you would get c on its own and find out what c is. The answer would be 5 = c.

20) Applying the order of operations on expressions with powers involving negative exponents and variable bases.

So, we all know that whenever you are doing the order of operations, you always have to use BEDMAS. BEDMAS is a specific order that you have to follow so that you don’t get anything wrong. BEDMAS (brackets, exponents, division or multiplication, addiction or subtraction) is extremely important to use in operations. As an example, we can use this operation to solve/simplify: (p⁻² x p⁶ x p⁻³)/(q⁷ x p⁻³). The first step is to simplify what would be on the top so with the power rule, you add (-2), 6, and (-3) (the exponents) together which equals a simple 1. Now, the equation would be (p¹)/(q⁷ x p⁻³). You cannot use the product rule for this one because it’s a division, not a multiplication, and so now you would use the quotient rule for p¹ and p⁻³ which is p¹⁻⁽⁻³⁾ which equals to p⁴, and now the fully simplified form would be p⁴/q⁷.

________________________________________________________________________________

That was quite a bit of information to take in but I feel like it’s a good learning opportunity for anyone and I hope you guys learned something new from the answers. I feel like since joining a new math class in honors, I have already learned quite a bit and I look forward to learning more things throughout the year. Anyways, thank you all for taking your time to read this and until next time. Thank you!

________________________________________________________________________________

Here is the core competencies/self-reflection.