Everything I Know About Exponents
1.Represent repeated multiplication with exponents:
5^3: 5 is the base, and 3 is the exponent. Multiply the base by itself however many times the exponent is. More examples would be:
Exponents are never the base multiplied by the exponent.
3. Demonstrate the difference between the exponent and the base by building models of a given power, such as 3^2 and 2^3.
The area of the square is L x W, 3cm is the length and the width, = 3cm x 3cm = 9cm squared. The volume of a cube is L x W x H, which is 2cm for all of them, that equals 2 x 2 x 2 = 8cm cubed.
3 squared is 3 x 3, whereas 2 cubed is 2 x 2 x 2. 
5.Evaluate powers with integral bases (excluding base 0) and whole number exponents.
(-2)^3 is using an integral base to the power of 3, and since that 3 is an odd number the question will be negative, this rule applies if their are brackets around the base, which will be explained in a following question. The answer would be -8. If the question is -2^4 than the answer will still be negative because the brackets were taken away and the it is 2 to the power of 4, which is 16, multiplied by -1. Using whole numbers is just using repeated multiplication.
7.Explain the exponent laws for multiplying and dividing powers with the same base.
If the question is 3 to the power of 5 x 3 to the power of 2 add the exponents and keep the base.= 3^7
If the question is the opposite where 3^5 is being divided by 3^2, then keep the base, and subtract the exponents = 3^3
9.Explain the law for powers with an exponent of zero.The exponent of 0 will always equal 1, no matter what number is the base, unless the base = 0.
11.Explain the law for powers with negative exponents.
If an exponent is left a negative then the answer is not considered simpified. For the exponent to not be negative then change the base to its reciprocal, and bring the exponent with it. Now that the exponent is on the bottom of the fraction, it is no longer negative. if there is a negative exponent in the denominator position, do the opposite, bring the negative exponents up with its base, to the numerator position. Only the base is moved to the denominator positsion, if there is a coefficiant, (a number/variable that is being multiplied by the exponent) then leave that at the top.
13.I can apply the exponent laws to powers with both integral and variable bases.
x^4 * x^3, keep the base, (x) then add the exponents. = x^7
(-3)^5 divided by (-3)^3 , keep the base (-3) subtract the exponents.
= (-3)^2 = 9
15.Use the order of operations on expressions with powers.
use BEDMAS, (brackets), Exponents^, Division/, Multiplication*,Addition+, Subtraction- To solve.
17.Identify the error in applying the order of operations in an incorrect solution.
Very well done. This helped me understand what you are doing in math and that you understand it as well. It made sense to me, (as an outsider), and I think you did a good job explaining with the pictures. Nice job, hope to see more in the future!
– Dave Butcher (father)
I think that it’s clear you put a lot of work and effort into this project and I really liked how concise and straightforward all your explanations were. It was easy to follow your examples because you added color and highlights and helped to distinguish the powers. Some of your examples definitely helped my understanding of exponents, especially the law for negative exponents. I think one thing you could improve is on number 19 maybe you could propose a question and answer it along with the formula for solving, although what you already have is good I think that if you are able to more clearly display what a growth problem is you will be able to have your reader leave with a stronger understanding. Another thing that you might want to change is the number of examples your providing, I feel like if you were to add some additional examples it would add to the depth and amount of knowledge the person reading your post takes away. Specifically, you might want to add some more examples to 1, 11, 13, 17 and 19. Although you listed many good things and provided quality examples seeing it done more than once would definitely help me. Overall your project was very informing, and I really enjoyed reading it.
Thanks for the comment! It was much appreciated.
Of course, and I’m really sorry for the late comment but I found some errors in your post that you might want to change. I liked how you added more examples however, in question 1 you wrote ‘more example’ so you might want to change that to something like ‘another example’ or ‘more examples’. In question 9 your photo said that 2/8=4, but 2/8=1/4 or 0.25. I think that you meant to write 8/2=4. In question 11 you said ‘all negative exponents are unsolvable’ and then you proceeded to explain how to solve them so you may want to change some of the wording there. Also in question 11, in your photo, you said that a4^-2=a 1/16 but you forgot that the ‘a’ would be on top to start, in fact you explained this when you were talking about how to reduce negative fractions. In your photo in question 17, you wrote that x^5 / 1/x^3 = x^2, but I think you forgot to reciprocal while dividing which resulted in the wrong answer because x^5 / 1/x^3 = x^8. Other than those minor things I really think that your post is very well done.