Everything I Know About Exponents
Describe how powers represent repeated multiplication
The power represents how many times you multiply the base by itself. For example : 2 to the power of 3 = 2 x 2 x 2 = because the base is 2. The answer will be 8. Therefore, powers represent repeated multiplication because the power shows how many times you have to multiply the base number by itself.
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.
Because the powers show how many times the number has to multiply by itself, in repeated multiplication, it will look like this.
Just because the powers and the bases are interchanged, doesn’t mean the answer will be the same.
Explain the role of parentheses in powers by evaluating a given set of powers such as (-2)4, (-24), -24
When the base is in brackets and the power is not, multiply that number exactly in the brackets by whatever the power shows. When the power is also in brackets and the base is a negative number, the negative symbol is the coefficient. For example, -n3 = -1 x n x n x n. For -24, it would be the same answer as the one before.
Explain the exponent laws for raising a product and quotient to an exponent.
Product law : When there is multiplication between exponents and the bases are the same, we can leave the bases and we add the powers by themselves. In a longer version, it is not simply adding them together. When the example is the one shown below, it’s basically (x . x . x . x . x) . (x . x . x). After that, you would add all the x‘s together and get a result of x8. If we did it the longer way:
For example: 34 x 32
= (3x3x3x3) x (3×3)
=(4 threes and 2 threes)
=6 threes
=36
Example:
3(x4) x 2(x2)
=(3 x 2) x (x4+2)
=6x6
Quotient law : When it is dividing and the bases are the same, you subtract the powers by the first one and the next one. Again, for this you would do the same thing to find out why it’s 5 – 3. (x . x . x . x . x) ÷ (x . x . x). Then you would do subtraction 5 – 3 and get a result of x2. If we gave an example with a coefficient, it would look like this:
(15x5) ÷ (5x3)
(When evaluating the coefficient, you would evaluate it like normal division ex. 15 ÷ 5 whilst you do the quotient law on the x’s)
= (15 ÷ 5) (x5-3)
= 3x2
Use patterns to show that a power with an exponent of zero is equal to one.
The best way to explain it is when we use the quotient law. When it is x3/x3 it will equal x0 because it is x3-3 so it will equal x0. If x was 2 then it will be 8/8 but 8/8 equals 1 because there is one 8 in 8. Therefore, any exponent that has a power of 0 will equal to 1. Also x cannot equal 0.
Use patterns to explain the negative exponent law.
Whenever there is a negative exponent, we do not do the same that we would normally do with positive exponents but instead we make it into a fraction. If it was
x–n, we would try to get rid of the (-) sign so that we could actually solve the equation. To make it into a fraction, we discard the negative sign and put “1” as the numerator of our fraction. Just like the one shown on paper below.
The reason why we make it into a fraction is not simply getting a one. We’re actually finding the reciprocal of the negative sign which is 1. the negative sign is basically a coefficient of n.
I can identify the error in a simplification of an expression involving powers.
For most of the time, I can identify what mistakes I’ve done or what others done when they ask to review their quizzes or tests. When I have to identify the mistakes regarding the power law, product law, or the quotient law, it’s easy for me because it’s just the matter of adding, subtracting and multiplying which is a basic skill that we learn in elementary school. But when it’s the answer of a long and big question, I lack a bit of confidence because I have to go through all the steps again to verify. Also, sometimes I might make some mistakes too because no one is that perfect and I’m not a math teacher also.
For example, in a quiz in exponents that we got back, we had to correct our questions that were wrong but I never asked the teacher of what I got wrong. I knew right away what I got wrong and did the corrections without any struggle.
The question was:
x5 ● x3. As my wrong answer, I subtracted them so I got x2 instead of x8. Cool thing is, I knew exactly what I got wrong and didn’t need help unless I got something wrong that I really don’t know about.
Determine the sum and difference of two powers.
For example, we have this equation : 32 + 23
Using BEDMAS, we would first solve the exponents. So, it would be 9 + 8, then we would do the addition so it would equal 17.
For 32 – 23
It would be 9 – 8 so the answer would be 1
Use powers to solve problems (measurement problems)
If we are trying to find the area of a rectangle and the shorter side equals x4 and the longer side equals x7, we would use the product law since we are trying to find the area.
Applying the order of operations on expressions with powers involving negative exponents and variable bases.
When it is a negative power, we cannot do the same procedures as we would do for normal positive powers. When it is a negative power, we get the reciprocal of the power which in this case, would be 1/3 because that is the reciprocal of 3. Then we would keep the x with the power and it would turn into a fraction like the image above.
Another one for practice is this but it is a bit more hard.
1: You simplify the negative exponent on the top and move the x3 to the denominator, you use the product law to the ones in the brackets
2: Still on the denominator, use the power law to find a single power
3: Use product law to multiply the two powers on the denominator
4: Evaluate the exponent
Core Competencies Reflection
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Hello Gideon, great work on your post! I really like the way you did your explanations. You addressed all the learning outcomes that you were required to. I found that your explanations are very simple and easy to understand. I did not see any mathematical errors in your work which is very good to see. Overall, your work is done very well, your examples and explanations are very clear and simple for anyone to understand. One thing I would add would probably to make it easier to understand for people who aren’t visual learners and mostly learn from reading text but other than that I feel that you did great!
I can see that you did a lot of work and I can see your effort put into this project. There is so much writing though and I hope that you could be more clear next time.
Gideon’s mom
Nice post Gideon. I like how you addressed each problem, and they were all easy to understand. when I read through all of the learning outcomes I understood all of them and didn’t find any incorrect math. I like how you used pictures for your explanations. I think you there was some unnecessary words in your post that you good remove. Otherwise, nice job!