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) (x^5-3)

= 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 x^3-3 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