Everything I know about exponents

Prescribed Learning Outcomes for Exponents:

1) Represent repeated multiplication with exponents

When representing repeated multiplication with exponents it is important to know that any number with a positive exponent, is the base multiplied by itself as many times as the exponent shows. Ex. $2^3$ is the same as $2\cdot2\cdot2$ Remember NEVER multiply the base and the exponent together.

2) Describe how powers represent repeated multiplication

Powers represent repeated multiplication because when you are evaluating powers, the power always represents how many times you multiply the base. ex. $2^3$ means the base (2) is being multiplied by itself, three times.

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 models can be the area of a square or the volume of a cube because Length times Width is area of a square (2 side lengths= $x^2$ ) and Length times Width times Height is the volume of a cube.(3 side lengths= $x^3$ ) $2^3$ can be represented as the volume of a cube because of the 3 side lengths, $2\cdot2\cdot2$ (cm) the answer would be $8cm^3$ and $3^2$ can be represented as the area of a square because of the 2 side lengths, $3\cdot3$ (cm) the answer would be $9cm^2$ .

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 $3^2$ .

$2^3$ = $2\cdot2\cdot2$ =8

$3^2$ = $3\cdot3$ =9

Don’t forget that we don’t multiply the base times the exponent, if we were to do that to these questions the answers would turn out the same but when we did the proper work the answers turned out to be different.

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

All bases except zero can be put to whole number exponents. Ex. $1^2$ = $1\cdot1$ = 1 works. But on the other hand $0^2$ does NOT work. $0^2$ = 0

6) Explain the role of parentheses in powers by evaluating a given set of powers such as  $(-2)^4$ , $(-2^4)$ and $-2^4$

Depending on the place of your parentheses the answer will be changed completely. If the exponent is outside the parentheses that means at the base is negative in example number one. If the exponent is in the parentheses that means that there is a coefficient, in example be 2. The coefficient is -1. And same with example number 3. Down below are the different ways you can put in your parentheses;

Ex.1 $(-2)^4$ = (-2)(-2)(-2)(-2) = 16

Ex.2 $(-2^4)$ = (-1)(2)(2)(2)(2) = -16

Ex.3 – $2^4$ = (-1)(2)(2)(2)(2) = -16

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

When multiplying powers of the same base (Product Law): keep the base the same and add the exponents.

Ex.

$3^2\cdot3^2$

= $3^{2+2}$

= $3^4$

When multiplying powers of the same base with coefficients (product law with coefficients): keep the base the same, add the exponents and multiply the coefficients.

Ex.

2( $2^2$ ) $\cdot4$ ( $2^3$ )

= $2\cdot4$ ( $2^{2+3})$

= 8( $2^5)$

= $8\cdot32$

= 256

When dividing powers of the same base (Quotient law): keep the base the same and subtract the exponents.

Ex.

$3^4\div3^2$

= $3^{4-2}$

= $3^2$

= 9

When dividing powers of the same base with coefficients ( quotient law with coefficients): keep the base the same, subtract the exponents and divide the coefficients.

Ex.

$15(2^4)\div3(2^2)$

= 5 $(2^{4-2})$

= 5 $(2^2)$

= 20

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

When raising a product or quotient to an exponent (power law): keep the base the same and multiply the exponents. Keep in mind that the outside exponent is multiplied to everything inside the parentheses.

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

When evaluating powers within exponent of zero the answer will always be one,but the base canNOT be zero.

Zero law is $x^0$ = 1 x cannot equal 0

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

There are many different patterns to show that a power with an exponent of zero equals one here are some patterns;

11) Explain the law for powers with negative exponents.

When evaluating powers with negative exponents (negative exponents law): reciprocal the power so it becomes the denominator and make the exponent positive.

12) Use patterns to explain the negative exponent law.

The pattern below shows that as the exponent decreases by one the answer is divided by two. The negative exponent law follows the pattern of all the laws including the zero law.

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

When you apply exponent laws to either veritable bases or integral bases the laws will always stay the same. If there any coefficients, use the same laws.

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

I know I can identify areas in a simplification of an expression involving powers because I know how to apply each law and the rules of each law. And I think it is very important to understand each and every step of every law incase you need to teach some one or its on a test. (Which it probably will be on)

Ex. $(2^2)^3$

= $2^{2+3}$

= $2^5$

= 32

Do not add exponents, multiply them.

$(2^2)^3$

= $2^{2\cdot3}$

= $2^6$

= 64

15) Use the order of operations on expressions with powers.

You use order of operations (BEDMAS) when you are adding or subtracting exponents because there is no law for them, you also use order of operations when the bases of multiplication and division expressions aren’t the same.

Ex. $2^4+3^2$ = 16+9 = 27

16) Determine the sum and difference of two powers.

When you were determining the sum or difference of two powers, just use BEDMAS.

Ex. $2^4$ + $3^3$ = 16+27 = 43

Ex. $3^3$ – $2^2$ = 9-4 = 5

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

When trying to identify an error in applying order of operations the biggest mistake is not going from left to right in order BEDMAS. If you do not go from left to right and you could end up with a totally different answer. You always need to double check your work for silly errors like these.

18) Use powers to solve problems (measurement problems)

When using powers to solve problems, you’re trying to find the area of the shaded part. To find the shaded area of the shape you must find the area of the non shaded part and the area of both shapes put together. And subtract the area of the white part from the area of both shapes. Or use Pythagorus to find a side length then calculate the area of the shaded part. And if the white part was a circle and was inscribed in a square you would use the formula pi R squared and then subtract that from the area of the square.

Ex. For the first picture you would do: $3cm\cdot3cm$ = $9cm^2$ and $2cm\cdot2cm$ = $4cm^2$ . Then you would subtract $9cm^2$ from $4cm^2$ = $5cm^2$ . So the area of the shaded part is $5cm^2$ .

Ex. For the second picture you would take the same steps but instead of just calculating only 1 white part calculate the area of ALL the white parts; $12cm\cdot12cm$ = $144cm^2$ and $3cm\cdot3cm$ = $9cm^2$ Then you do $9cm^2\cdot4$ = $36cm^2$ Because there are 4 of the same white squares. Then $144cm^2$ – $36cm^2$ = $108cm^2$ so the area of the shaded part is $108cm^2$ .

Ex. For the third picture you need to find the side length of the square, and to do that you must use Pythagorean Theorem; $a^2+b^2=c^2$ You need to find what X is. To do that you have to isolate the X and you will find your answer but you will need to find the square root of it since its $x^2$ and X could be positive or negative but since X has to be greater than zero that means that X has to be positive. In this case +12. So the area of the shaded part is $12m\cdot12m=144m^2$ .

19) Use powers to solve problems (growth problems)

When you’re using powers to solve growth problems, you need to know how much something is growing by in a certain amount of time, in any unit. In the example below we are tripling every hour. And you also need to know what number you are starting off with. In the example below we are starting with 50 candies.

The steps are; how much you start off with, multiplied by how many times it grows (ex.triple=3,double=2) and the exponent would be in how much time(1hour=1, 2hours=2)

My bag of candy triples every hour. There are 50 candies now. How many will there be after each amount of time?

A.) 1 hour= $50\cdot3^1=150$

B.) 3 hours= $50\cdot3^3=1350$

C.) 5 hours= $50\cdot3^5=12150$

D.) x hours= $50\cdot3^x$

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

When applying order of operations on expressions with powers involving negative exponents and variable bases it is important not to skip any steps. The first thing you should do, is deal with your coefficients, then complete any laws that apply, and if negative exponents are left, reciprocal the power to get rid of the negative exponent.

And I also know:

Anything to the power of one always equals the base.

One raised to any exponent is always going to be one.

A coefficient is the shortcut for repeated addition.

4 thoughts on “Everything I know about exponents”

Sabrina on November 11, 2016

Great work Karina! I really liked how detailed your post is and that there is so much valuable information. The pictures add a great affect to the presentation of your post. Overall it looks excellent!

shannonb2016 on November 12, 2016

I know I’m supposed to comment on things to work on but I can’t find anything. I read the whole thing and I couldn’t find any mistakes, everything was easy to understand and all the learning outcomes are there. I really like how you kept everything so organized. One thing for improvement is that in your examples they look squished, make them more spaced out if possible, it all kinda mixes together. Great job!

- karinab2016 on November 13, 2016
  
  Hey Shannon, thank you for the kind comments. I took your advice and actually did space out some examples. Thanks again!
  
thubbard on November 16, 2016

Thanks Karina, Please check your Onenote for specific feedback.

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Everything I know about exponents

4 thoughts on “Everything I know about exponents”

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