Everything i know about exponents

alext2016Math 9

1) Represent repeated multiplication with exponents

To represent repeated multiplication with exponents you tale the multiplication ex. (2x2x2x2) you take the number and put it as the base and the amount the number is repeated is the exponent. so my example would be 2^4

2) Describe how powers represent repeated multiplication

Powers represent repeated multiplication by telling you how many times the base is repeated in the mutiplication. For example 2^3 is equal to 2x2x2

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 model is a cube. lets say the cube has a side length of 4. so to find the volume of said cube we would have to do 4x4x4 or 4^3. The difference is that the exponent is the number of times the side length is repeated and the base is the side length.

4) Demonstrate the difference between two given powers in which the exponent and the base are interchanged by using repeated multiplication, such as 2^3 and 3^2

2^3=2x2x2=8

3^2=3×3=9

There would be a difference because we dont multiply the base by the exponent instead we multiply the base by itself the number of times that the exponent is.

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

You cannot do powers with 0 as the base. To evaluate powers with intergral bases and whole number exponents you just multiply the base by itself however many times the exponent tells you to. For example 5^3 = 5x5x5. since the exponent is three you must multiply the 5 by itself 3 times.

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

If the exponent is outside of the bracket you multiply whatever is inside the bracket by itself however many times the exponenet tells you to. But if the exponent is inside and the base is negative you multiply the base by itself however many times the exponent tells you to then multiply that by -1. and if there are no brackets you do the same as if the exponent was inside the bracket because brackets make no difference in this situation.

(-2)^4=(-2)(-2)(-2)(-2)

(-2^4)=(-1)(-2)(-2)(-2)(-2)

-2^4=(-1)(-2)(-2)(-2)(-2)

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

If youre multiplying exponents with the same base that is called the product law. you do it by keeping the base the same and adding its exponents together. 5^2x5^3=5^5

If youre dividing exponents with the same base that is called the quotient law. you do it by keeping the same base and subtracting the exponents. 5^3\div5^1=5^2

If youre multiplying exponents with the same base with coefficients this is called the product law with coefficients. you do it by keeping the base the same, adding the exponents and multiplying the coefficients. 2(5^2)x3(5^4)=6(5^6)

If youre dividing exponents with the same base with coefficients this is called the quotient law with coefficients. You do it by keeping the base the same, subracting the exponents and dividing the coefficients. 4(5^4)\div{2(5^2)}=2(5^2)

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

If you are raising a product and a quotient to an exponent it is called the power law. you do this by keeping the base and multiplying the outer exponent by the exponents inside the brackets.

{(2^1\cdot2^2)^2}={2^2\cdot2^4}

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

When the exponent is 0 the answer will always be one. 10000^0=1

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

We can just use BEDMAS and the quotient law. For example:

3^3\div3^3=27\div27=1

3^3\div3^3=3^{3-3}=3^0

and since both the answers should be the same, that makes 3^0=1

11) Explain the law for powers with negative exponents

If you have a negative exponent you need to use the negative exponents law. you do it by recipricalling it then evaluating to make the exponent positive.

3^{-2}= \frac{1}{3^2}=\frac{1}{9}

12) Use patterns to explain the negative exponent law

2^2=4

2^1=2

2^0=1

2^{-1}=\frac{1}{2}

2^{-2}=\frac{1}{4}

the patters continues to divide by 2

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

You use the same laws for variable bases

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

I can identify errors because i know the laws and how to do them.

4^5x4^3=4^{5\cdot3} that is wrong because in the product law you add the exponents not multiply them. 4^5x4^3=4^{5+3}

15) Use the order of operations on expressions with powers

If you are subracting or adding exponents you should use BEDMAS because there is no law for adding or subracting them. 5^2+2^1= 25+2=27

16) Determine the sum and difference of two powers

To determine the sum and difference of two powers we use BEDMAS

2^2+2^1=4+2=6

2^3–2^2=8-4=12

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

2^2-2+4=4-6=-2 this is wrong because they added but they should have subracted because you should do things form left to right. the correct answer would be: 2^2-2+4=4-2+4=2+4=6. You can see that the answers are different.

18) Use powers to solve problems (measurement problems)

If youre trying to find the area of a shaded part of a square. you find the side lenth of the big square then find the side length of the small square which is un shaded. lets say the side length of the big square is 3 and the small square is 2. first, you find the area by putting the side lengths to the power of 2. 3^2 and 2^2. then you do the math and find the area. the big sqaures area would be 9 units squared and the small squares area would be 4 unit squared. then you subract the small squares area from the big squares area: 9-4=5. so the area of the shaded part would be 5 units squared.

19) Use powers to solve problems (growth problems)

If youre solving a growth problem with powers you need to know how much you start off with and how long it grows for. how much you start with would be your coefficient and how much it grows would be your exponent. You also need to know the growth rate and that would be your base. Example: I have 10 dollars and the amount of money i have doubles per hour. how much money would i have in 2 hours and how much in 3 hours?

2 hours: 10(2^2)=10×4=40

3 hours: 10(2^3)=10×8=80

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

All you need to do is use BEDMAS and follow the laws.

\frac{2^{-2}}{{2a}^{-2}}=\frac{{2a}^2}{2^2}=\frac{4a^2}{4}

2 thoughts on “Everything i know about exponents”

  1. Hi Alex, You addressed all of the mathematical concepts that were asked. I was able to understand all all of your explanations, I couldn’t find any mathematical errors but for #3 even though you explained the way that 2^3 works as a model saying that it’s a cube you forgot to mention that 3^2 is different because it’s a square. I really liked the way you explained #16 you showed clear steps on how to find the sum and difference of exponents. I liked the way you explained #10 because the reason that the zero law exists is one of the hardest concepts for me to understand and that was a very clear way of explaining. I also liked the way you explained #19 it never occurred to me explaining the growth problem through money but it makes sense how you do it! For next time I would encourage that you use more visuals so that the reader can understand questions like #3 and #18 better because even though you explained the way the square would look it would be nice to have a picture of the square on you’re blog. Also for some of the questions ex #13 maybe go into more detail because some readers may not know what the laws for integral bases are therefore they wouldn’t know how to execute questions with variable bases. Otherwise great job!

    Reply

  2. You have some clear explanations of the exponent laws with good examples. It looks like you have a good grasp of these concepts. I also like how you have included mathematical symbols in your post. I have to agree with Ashiana that some visuals would help get your message across even better. Great work here!

    Reply

Leave a Reply

Your email address will not be published. Required fields are marked *