Category: Math 9

As the name of my post suggests, these are all the things I know and have learned about our unit on exponents and I am looking forward to sharing this with you all. Any feed back would be greatly appreciated!! Don’t be afraid to mention any parts that you think I need to work on, I would love to fix my mistakes and try to make this perfect! Thanks! 

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1.Represent repeated multiplication with exponents

When you have a number like 2³, all it means is there are 3 tows being multiplied together 

Ex:  2 x 2 x 2= 4 x 2= 8. 2x2x2 is also represented as and should be represented as 2³

This can also be used in the other way around in the sense that instead of writing 3x3x3x3, you can simply just say its 3⁴. The 3 is represent the base which is the number being multiplied and the 4 is the exponent which is the amount of 3’s is being multiplied by each other.

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3. Demonstrate the difference between the exponent and the base by building models of a given power, such as 2³ and 3²

For example, when you have a number like 3², the three represents the number being multiplied, the base. Where as the exponent 2, is how many of the base there is going to be multiplied.

So, 3²= 3×3=9. The same thing goes for 2³.

2³= 2x2x2=8

If you were to have a coefficient involved into an equation, make sure to check if the exponent effects the coefficient and the base or just one. Such as:

2x³= 2x³

(2x)³  = 8x³

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

When evaluating powers with integral bases, it means positive and negative integers. The rule of the number of negatives determining if the answer is negative or not applies a lot here so make sure you pay attention to that because its always going to get you.

For Example, (-2)⁴ = 16 HOWEVER (-2⁴) does not equal 16 it equals -16 because the exponent is on the four which is acting as the base and the negative is acting as a coefficient. It would look like this: -1x2x2x2x2

When you have negative coefficients involved, it would be the exact same: (-2x)³= -8x³ , 

-2x³= -2x³

With positive integers it would be the exact same as the other examples but obviously without the negative.

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

When you have 2 numbers with the same base and you are dividing, you always keep the base and SUBTRACT the exponents. If there are any coefficients, divide them accordingly: 2³ Divided by 2² = 2¹ because you are doing 2^3-2 . This law is called the Quotient Law

For example, 3² divided by 3¹= Subtract the exponents: 2-1 and keep the base. Final answer is 3¹ because 3^2-1

When you have 2 numbers with the same base and you are multiplying, you always keep the base and ADD the exponents. If there are any coefficients, multiply them accordingly: 2³ x  2² = 2⁵ because you are doing 2³⁺² . The law is called the Product Law. Another way it can be explained is 2³  means 2x2x2 and 2² means 2×2. When you multiply them it will look like (2x2x2)(2×2) which means there are 5 twos which is……. 2⁵

2³ X 2⁵= add the exponents: 3 + 5 and keep the base of 2. Final answer is 2⁸

With coefficients: 2x³ X 4x⁴ = 8x⁷

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

When any number has an exponent of zero, the answer is always 1 (except when its 0⁰, its equals zero) because when we take a number like 2² it equals 4 then 2¹ equals 2. We can see a pattern of the number being cut in half every time so we know the next one will always be 1. 2⁰ equals 1.

Another way of wording it is that we would divide the product by its original base. If 2² = 4 then we divide that by 2 and we get 2 which is equal to 2¹, divide by 2 again and we get 1 which is equal to 2⁰.

Ex: 2² =4  2¹ =2  2⁰=1  2^-1= 1/2

Ex: 3² = 9  3¹ = 3  3⁰ = 1  3^-1 = 1/3

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11. Explain the law for powers with negative exponents.

Now for the situations where you find yourself dealing with negative exponents, the rule that is the easiest to follow is that you just put the base in its reciprocal form then change the exponent to a positive and solve. The reason why you put it in its reciprocal form is because if I were to put the entirety of 1/2 into its reciprocal, I would get a complex fraction which is basically a fraction on top of a fraction, hence the name complex fraction. We don’t want a complex fraction so we would just flip 1/2 to 2/1 and apply the exponent like normal.

Ex: 2^-1 equals (1/2)¹ = 1/2 or another example is (1/2)^-1 = 2¹ =2

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

Product law: As stated before, when multiplying exponents, you will always keep the base and add the exponents IF the base is the same. If there are any coefficients, multiply them.

Examples: 2³ X 2⁹ = 2¹²,  190⁴ X 190¹² = 190¹⁶

3(2¹) x 2(2³) = 6(2⁴)

With variables: 3x³ X 6x⁶= 18x⁹

Quotient law: ALSO stated before, when dividing exponents, you must keep the base and subtract the exponents IF THE BASE IS THE SAME. If there are any coefficients divide them.

Examples: 5⁶ divided by 5² = 5⁴,    4(6⁷) divided by 2(6⁴) = 2(6³)

With variables, its basically the same: 5m⁷ divided by 5m⁵= m² because, 5 divided by 5 is 1 and

M^7-5=2

Power Law: The one law I personally hate. This law is when there is an exponent being applied to an exponent. Sounds confusing right? I thought that at first but its not that bad. When you get a question like that, you are just going to multiply the exponents, keep the base but don’t forget about those pesky coefficients! You’ll need to apply the exponent to those too.

Examples: (2³)²= 2⁶,     (3×4²)²= 9×4⁴ or 9(4⁴)

The power law with coefficients behaves the exactly the same way: 2(n⁵)⁵= 2n^25,       (6c³)²= 36c⁶

Negative exponents: For this one, your going to have to reciprocal what ever base the negative exponent is on before you can apply that bad boy on said base. As I stated in the last question, make sure you don’t accidentally make a complex fraction because it WILL get complex. Other than that, its simple as pie.

Examples: (2)^-2 = (1/2)^-2 = 4,    (1/3)^-3= (3/1)³= 27/1 or 27

Zero Law: Hands down anyone’s favorite, IT’S ONE. That’s it. No matter how ginormous the number is, if its got a 0 for an exponent its- and you guessed it, ONE. All you do is take one product of the base having an exponent of whatever and just divide it by its base until you get to the exponent 0 where you will see that it always equals 1 no matter what. Another way I like to remember it was, exponent zero? ONE.

Example: 3³= 27  Divide by base (3) = 9 , 9 is equivalent to 3²  Divide by 3= 3 , 3 is equivalent to 3¹ Divide by 3 = 1

Examples: 678(2467)⁰= 678(1) =678    4(278⁰)= 4(1) =4

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15. Use the order of operations on expressions with powers

The order of operations still applies when dealing with exponents always. PEDMAS always stands and can still cause you to make some mistakes if you don’t follow it.

To identify when you could use exponents first can be when multiplication is being done inside the parentheses: 5(3×4)² = 5(3²x4²) =5(144)= 720. It doesn’t change the answer because both ways, you would get the same result

However, when there is addition or subtraction in the brackets, you must evaluate the addition or subtraction first before applying the exponent because the answer WILL NOT be the same: 9(4-3)³= 9(1)³=9            25(5-2)² = 25(3)³ = 25(27) =675

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

Always make sure to double check your equations and your answers, I make a ton of very small and stupid mistakes. To give a little insight in what I mean, here is an example:

 This is wrong 2(6+7)²= 2(6²+7²) Do not do that. The problem with that is when there is addition or subtraction inside brackets, always evaluate that first before the exponents. The right way is 2(6+7)² =2(13)²= 2(169)= 338

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19. Use powers to solve problems (growth problems)

When given a question such as: The cells doubled every hour, how many are there after 2 hours when you start with 3.

we know we have 3 from the very start and every hour it triples. So, to put this in a equation, we can say at 0 hours we have 3, at 1 hour it triples so then it would be 9 then at 2 hours, 27

In exponent for we can write it as: 3(3ⁿ). The 3 on the outside is representing the amount we start with, the 3 on the inside representing the amount it goes up every hour and the exponent N represents the hour. If N were 1, then it would be 3 x 3¹ which is 9 and so fourth……

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Conclusion

 I am sure all your brains are fried but hey, you learn something new everyday or in my case every day 2. I do really hope you have learned something you didn’t know already! In all, I feel that I have learned so much more about exponents and math in general in this class. Its really taught me a lot of new things and concepts that I did not have a lot of knowledge on before. I am really looking forward to learning even more!

 

My partner Daejung did the even Learning outcomes to this project, the link to his blog is: http://myriverside.sd43.bc.ca/daejungc2019/2019/11/08/everything-i-know-about-exponents/

CC