1. Represent repeated multiplication with exponents

All repeated multiplication can be turned into powers by writing the number with an exponent of how many times it repeats. For example: 2 * 2 * 2 * 2 = 2^{4} because there are 4 two’s multiplying by each other to get an exponent of 4.

2.Describe how powers represent repeated multiplication

A power represents repeated multiplication by having the base, showing the number being multiplied and the exponent, showing how many times the base is multiplied by. In this image the 4 is the exponent showing how many times the two is multiplied.

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 difference between the exponent and the base are that the base is the number being multiplied and the exponent is how many times the number gets multiplied. In this picture it shows that 2^{3} is a three dimensional cube, and the exponent of 3 to the side length calculates volume and 3^{2} is a two dimensional square, and the exponent of 2 to the side length calculates surface area.

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}.

When powers are interchanged meaning the base and exponent are flipped it changes the power completely. The number being multiplied and the amount of times it is being multiplied is changed.

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

When there is a positive base and exponent the answer will always be positive. When the base is negative, and the exponent Is positive it is different. If there is an even number exponent and a negative base the answer will be positive but when the base is negative, and the exponent is an odd number then the answer is going to be negative.

6. Explain the role of parenthesis in powers by evaluating a given set of powers such as (-2)^{4}, (-2^{4}), and -2^{4}.

Parenthesis can change the exponent completely. When there are negative bases the parenthesis needs to be put around it for the base to be recognized as a negative base instead of a positive base multiplied by -1. In this image, it shows how if the base isn’t in brackets separated from the exponent, the negative turns into -1 then 2s multiplied instead of -2s multiplied.

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

When multiplying and dividing a power with the same base there are laws to simplify the question easier. When multiplying powers with the same base there is a product rule where you can add the exponents together to get one power. When doing this you also multiply the coefficients together. Also, when dividing powers with the same base there is a quotient rule where you can subtract the second exponent from the first one to get one power.

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

When raising a product and quotient to an exponent you multiply all the exponents inside the brackets by the exponent on the outside of the bracket. When doing this you also add the exponent on the outside of the brackets to the bases on the inside of the brackets.

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

When an exponent is to the power of 0 it always equals one. So, 27, 3, 9, and anything else to the power of 0 equals 1. This works this way because any number divided by itself is equal to 1, except doesn’t equal 1, because 0 divided by 0 = undefined.

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

Every time the exponent gets one smaller you divide the previous answer by 3 and this shows that anything to the power of 0 is 1.

11. Explain the law for powers with negative exponents.

When there is a negative power you flip the base to make the power positive.

12. Use patterns to explain the negative exponent law.

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

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

When simplifying expressions with exponents, many things could go wrong. Some things that could go wrong are multiplying a base to exponent instead of correctly simplifying the power, adding exponents on addition questions, or subtracting exponents in questions involving subtraction.

15. Use the order of operations in expressions to powers.

When simplifying expressions with the order of operations, I use BEDMAS. This is the steps to take to solve a question. Starting with brackets, then exponents, division and multiplication, then finally addition and subtraction.

16. Determine the sum and difference of two powers.

When finding the sum and difference of two powers there is no short cuts.

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

To identify errors in order of operations, I use BEDMAS. I check that the question is done with brackets and exponents first, division and multiplication second, and addition and subtraction last. In this question the mistake was missing the brackets. The correction is to do the inside of the brackets at the same time as the power to avoid a mistake.

18. Use the powers to solve problems (measurement problems)

When finding measurement problems, I use the power of two and three to find the surface area of shapes from their side lengths. Is use the power of two to find the surface area of two-dimensional shapes and the power of three to find the volume of three-dimensional shapes.

19. Use powers to solve problems (growth problems)

A colony of bacteria triples every hour. Right now, there are 100. How many are there after… on this problem the 100 * 3 is growth per hour, then the exponent at the end is for the number of hours it has been in the problem.

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

When there are negative powers, I flip the exponent to make it positive. In this question the first step to do is the exponents because there are no brackets. When the exponents are negative they get flipped to the bottom of the question (like shown in the picture). Then all the exponents turn positive, so the only thing left to do is divide. After dividing everything is when the answer will be found out.

Parent Comment:

Wow! This is Grade 9 math?! I could keep up until the BEDMAS questions (#17). You explained the concepts clearly and gave good examples to illustrate your thinking. Well done, Enzo! I don’t remember negative exponents that well so although your explanations sound good, I don’t know whether they’re correct or not. Sorry, it’s been too long since I’ve needed to use those concepts in real life….

Overall, I thought that your project really showed your comprehension of the exponents unit. I thought that your visuals really enhanced the the examples and increased the understanding of the question. Your explanations, for the most part, where very clear and I like how you gave the definition of words that may be more complex (for example you explained what interchanged meant for #4). There are a few things you could change.

For #10 You could clarify. Make sure you clarify that any number divided by itself is one which is why this rule is plausible.

For #12 Your picture would be clearer if you added a color to differentiate the numbers.

For #17 I would recommend writing out the question in word form so it is clearer what your drawing represents.

For #20 This example could be clearer but it was still understandable.

I think that this project was well done. Good Job!