2- Describe how powers represent repeated multiplication

A power (exponent) is basically a quick way to do repeated multiplication. Instead of writing down something like 5 x 5 x 5, you can instead write 5³which is basically a way of saying you’re multiplying 5 three times. 5³ = 5 x 5 x 5

4- Demonstrate the difference between two given powers in which the exponent and the base are interchanged by using repeated multiplication, such as 2 and 5.

The difference between having 5² and 2⁵ is in the first one, you are multiplying 5 twice, while in 2⁵, you are multiplying 2 five times, this is how they look different and make a different answer:

5² = 5 x 5 = 25

2⁵ = 2 x 2 x 2 x 2 x 2 = 32

6- Explain the role of parentheses in powers by evaluating a given set of powers such as

Brackets can be used to change the way exponents will work, for example, having a number like -3² doesn’t mean that the “-” is a part of the exponent but it’s actually a coefficient, if a “-” is there, you do the exponent of the positive number first, then multiply by the coefficient -1 (e.g. -3² = -1 x 3 x 3 = -9). If the exponent is outside brackets and the base is in them, like (-3)², then the “-” becomes part of the exponent, as it is an exponent of everything in the bracket (e.g. (-3)² = (-3) x (-3) = 9)

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

For raising a product to exponent, we use the product rule, where if there’s two numbers with the same base, we add together the exponents (e.g. (a²)(a³)= a²⁺³ = a⁵). The quotient law works the same way, except subtracting instead of adding:

(a³)/(a²)= a^3-2 = a

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

A number other than zero to the power of zero will always equal one because it’s even, so the number will be positive and when you lower a power by 1, it’s like dividing by the base

1 – divide by 1 every time	2 – divide by 2 every time	3 – divide by 3 every time	4 – divide by 4 every time
13 = 1	23 = 8	33 = 27	43 = 64
12 = 1	22 = 4	32 = 9	42 = 16
11 = 1	21 = 2	31 = 3	41 = 4
10 = 1	20 = 1	30 = 1	40 = 1

12- Use patterns to explain the negative exponent law.

Negative exponents aren’t negative numbers because the pattern in exponents is every time you raise the exponent by 1, you multiply by the base, so if you lower the exponent by 1, you divide by the base. If the exponent is lower than 0, we can also multiply the positive exponent by it’s reciprocal to find the same answer. e.g. 9² = 9 x 9 = 81, 9¹ = 9, 9⁰ = 9 / 9 = 1, 9^-1 = 1/9, 9^-2 = (1/9)/9 = 1/81, etc.

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

A common mistake in expressing simplification is adding a positive or negative when you don’t need one and order of operations, for example:

Wrong:                                                          Right:

(2³) ^2a (ab²)²                                                   (2³)^2a (ab²)²

= 2^6a a⁴ b⁴                                                       = 2^6a a² b⁴

Wrong                                                            Right:

-2⁴ (a²b³)²                                                        -2³ (a²b³)²

=-16a⁴b⁶                                                           =16a⁴b⁶

16- Determine the sum and difference of two powers.

You may want to break the expression down into smaller chunks if you think its easier for adding

a) 2⁶ + 2⁷ = 64 + 128 = 192

When subtracting a bigger number from a smaller number, you can instead subtract the smaller number from the larger one and multiply by -1 (x-y) = -(y-x)

b) 3³ – 3⁵ = 27 – 243 = -216

18- Use powers to solve problems (measurement problems)

Example: Taylor has 4 millimeter by 4 millimeter pieces of paper he plans to use for confetti at a party, is is arranged in 32 rows and columns with four missing in a corner so he knows how much he has made, find out how much confetti Taylor has made in millimeters

Because all of the numbers are a power of 2, you can change all the number to 2 and have exponent be whatever it takes to get back to the original number

4²mm ((32)² -4mm)

= (2⁴) ((2⁵)² – (2²))

= 2⁴ ((2¹⁰) – 2²)

= 16 (1024 – 4)

= 16 * 1020

= 16320mm

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

You always want to make sure you are doing everything inside the brackets first, then moving on to the exponents after, making use of any exponent laws, some common problems with using order of operation is that you will do something like add before dividing/multiplying or something similar.

e.g.      (9a^-4b⁷)^-2/(3³a²b^–3)^-1

=            (9^-2 a⁸ b^-14)/3^-3a^-2b³

=            (3³a¹⁰)/(9² b¹⁷)

=            (3³a¹⁰)/[(3² )² b¹⁷]

=            (3³a¹⁰)/(3⁴b¹⁷)

=            (3^-1a¹⁰)/b¹⁷

=            a¹⁰/(3b¹⁷)

Rich’s post: http://myriverside.sd43.bc.ca/richk2019/2019/11/11/everything-i-know-about-exponents/