Everything I Know About Exponents

1. Represent repeated multiplication with exponent

If you had a question that asked you what is equal to $3\cdot3\cdot3\cdot3$ that’s just a more difficult way of saying $3^4$ . its a form of $Exponents$ .

${x}\cdot{x}\cdot{x}\cdot{x}\cdot{x}$ = $x^5$

2.Describe how powers explain multiplication

Exponents are a faster form of repeated multiplication, just as multiplication is a faster form of repeated addition, some examples are

x+x+x+x = ${x}\cdot4$ , and exponents are

${x}\cdot{x}\cdot{x}\cdot{x}$ = $x^4$

so anything to an exponent is just the $Base$ multiplied the by itself the amount of time as represented by the exponent

3. Demonstrate the difference between the exponents and the base by building models of a power.

this is an example of

$2^3$ or 2 cubed

each side is 2 units long, so

its like saying $2\cdot2\cdot2$

and that’s the same as saying $2^3$

This is the flipped version of $2^3$ ,

$3^2$ . as you can see it has gone from a cube to a 2 dimensional shape which is another reason why the base and the exponent CAN NOT BE SWITCHED.

4. Demonstrate the difference two given powers in which the exponent and the base are interchanged by using repeated multiplication

The $Base$ and the $Exponent$ are two completely different things. The base is the bottem part of the $Power$ and the number that is being multiplied, the exponent is the top number in the power that is how many times you multiply the base.

THEY ARE NOT INTERCHANGEABLE

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

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

5. Explain the role of parentheses in powers

$Parentheses$ are very important when it comes to powers with negative bases. The difference between $-3^2$ , ${(-3)}^2$ and ${(}{-3}^2{)}$ important. $-3^2$ in repeated multiplication is ${-1}\cdot3\cdot3$ which is equal to -9, meanwhile ${(-3)}^2$ in repeated multiplication is ${(-3)}\cdot{(-3)}$ which is equal to 9. The third ${(}{-3}^2{)}$ is the same as $-3^2$ , just inside parentheses. if you get these things confused then you will end up with 2 completely different numbers. Parentheses also have their regular use of grouping things together as in P.E.D.M.A.S.

6. Evaluating exponents with integral bases and whole number exponents

When we are doing exponents with negative bases remember that if the exponent is even the answer will be positive and if it’s odd then the answer is negative. This rule only works for negative exponents inside parentheses. an example of this is:

${(-3)}^4$ = 81

${(-3)}^3$ = -27

${-3}^4$ = -81

${-3}^3$ = -27

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

When 2 powers have the same base, their exponents can be manipulated before you solve for the power. the 2 laws here are the $Product$ $law$ , and the $Quotient$ $law$ .

The product law is when both power’s bases are the same, you can add the exponents so ${x^6}\cdot{x^2}$ would be equal to $x^8$ .

The quotient law is when both powers have the same bases, their exponents can be subtracted ${x^4}\div{x^2}$ would be equal to $x^2$

8. explain the law of raising a power to an exponent

if you get a question that looks like this ${{(}x^3{)}}^4$ . this means you have to use the $Power$ $law$ . the power law says that if you an exponent outside parentheses containing a power, then you can keep the base and multiply the exponents, so ${{(}x^3{)}}^4$ would be equal to $x^{12}$ .

9. explain the exponent law for raising a quotient and a product to an exponent.

the quotient law and product law are above but here they are again,

The product law is when both power’s bases are the same, you can add the exponents and then multiply the $coefficients$ (if there are any) so ${x^6}\cdot{x^2}$ would be equal to $x^8$ .

The quotient law is when both powers have the same bases, their exponents can be subtracted, then divide the coefficients (if there are any) ${x^4}\div{x^2}$ would be equal to $x^2$

here is an example:

first you would add the exponents from 3 to the power of 4 and 3 to the power of 5,then subtract the exponents from 3 to the power of 7. this gives you $3^2$ . notice how the base didn’t change throughout the whole thing.

10. explain the law for powers with the exponent of zero

first any base with an exponent of 0 is equal to 1. this works because while $4^0$ = 1 may not make sense at first there is a way to prove it. if you use PEDMAS the in the expression

${6^2}\div{6^2}$ = ${36}\div{36}$ which is 1.

so if $6^0$ is the same thing, the logically its equal to 1

11. use patterns to explain the zero law

observe:

$2^5$ = 32

$2^4$ = 16

$2^3$ = 8

$2^2$ = 4

$2^1$ = 2

$2^0$ = 1

the pattern is that each time the numbers divide by 2 so if $2^1$ is equal to 1, then logically the next one down $2^0$ should be 1.

12. Identify the error in the simplification of an expression using exponents

$3^2$ = 6

the student who did this preformed $3\cdot2$ instead of $3\cdot3$

${3^2}\cdot{3^3}$ = $latex 3^6

instead of adding using the product law like you are supposed to, the student multiplied the exponents.

13.apply the laws of exponents

product law

${2^3}\cdot{2^4}$ = $2^7$

Quotient law

${5^7}\div{5^4}$ = $5^3$

power law

${({3^4})}^2$ = $3^8$

zero law

$5^0$ = 1

negative law

$3^{-1}$ = $\frac{1}{3^1}$

14. use order of operations on expressions with powers.

order of operations or PEDMAS is used on expressions where exponents are being added or subtracted as there is no way to do it in the form of a power

so in the expression: $3^2$ + $2^4$

you would do the exponents first turning it to 9+16 = 23

or now one that’s more complicated.

${5^5}\div{5^3}$ + ( $3^2$ – 4)+ ${2^3}\cdot{2}$

Answer:

you have to work it out in the order of operations.

first do the brackets, and inside use PEDMAS again and solve the exponent then subtract 4, making it 9-4=5, and now our question looks like

15. identify the sum and difference of 2 powers

you have to convert both to fractions first then add or subtract them

$4^2$ + $2^3$ = 16 + 8= 24

$3^3$ – $2^2$ = 27 – 4 = 23

16. identify the error in applying the order of operations with the incorrect solutions

a student answered this equation

$3^2$ – ${3^1}\cdot{3^2}$ = 1

what the student did is he added the exponents on the multiplication right, but then did 3 squared divided by 3 squared when he should have done 9-9 which is 0 not 1

17 and 19. use powers to and order of operations to solve problems(growth)

for the next 2 questions, a) is the powers problem and b) is the order of operations and powers part

write an expression using powers to solve the problem, then solve it.

a) if a single virus cell infected an animal, and it took 1h for that virus cell to corrupt a regular animal cell and multiply itself by 10, how many virus cells would there be in 12h?

Answer:

if a single virus cell infected an animal, and in 1 hour the 1 virus cell turned into 10, then that would be like saying $10^1$ , so if the exponent is the hours then the expression would just be $10^{12}$ and the number of virus cells in 12 hours would be: 1,000,000,000,000 virus cells

b) and if each virus cell can only live for 9h.

that means that any virus cell created past the 3rd hour would live, so the expression would be $10^{12}$ – $10^3$ and the amount of cells there are would be 999,999,999,000 virus cells inside the animal

18 and 20. use powers and order of operations to solve problems(measurement)

problem, write an expression using powers to find the area the question asks you for, then solve it.

$math-question$

a) find the total area of the outer square

it is a simple solution as $4^2$ would be the full area of the outer square, because exponents a

can be used to find the area of a square.

the are on the outer square is $latex {16cm}^2

Solution: because they’re both squares you have to subtract the area of the little square from the area of the big square.

so, the expression would be:

$4^2$ – $2^2$ which is

16-4= ${12cm}^2$

21. apply the exponent laws to with coefficients and variable bases

The 5 main laws of exponents are

– the product law

the formula to solving for the product law with coefficients is to have the base of the power stay the same, add the exponents together, and multiply the coefficients

Example:

a and b are coefficients, c and d are exponents and x is the base. the answer is:

${b}\cdot{a}$ $x^{c+d}$

– the quotient law

the quotient law is like the opposite of the exponent law, so you would keep the base, subtract the exponents and divide the coefficients

Example:

so the answer would be

$a\div{b}$ $latex x^{y-z}

– the power law

the power law is a simple one

keep the base and multiply the exponents the give the coefficient an exponent of its own, then solve.

Example

${(x^2)}^3$ = $x^{6}$

– the zero law

the zero law is the easiest of them all. any base with a coefficient of 0 is 1

Example:

$x^0$ =1

– the negative law

the negative law is probably the hardest of the 5, but there’s a trick to it as well. whenever an exponent is negative in a question, then you have to flip it with a fraction and the power that had the negative exponent now has a positive one, just on the other side of the fraction.

Example

$x^{-4}$ = $latex \frac{1}{x^4}

22. explain the negative exponent law and apply it to both integral and variable bases.

The $negative$ $exponent$ $law$ is the law that tells you how to deal with negative fractions. if we continue with the pattern from #11 then we see that

$2^2$ = 4

$2^1$ = 2

$2^0$ = 1

so if we continue with this pattern…

$2^{-1}$ = $\frac{1}{2}$

$2^{-2}$ = $\frac{1}{4}$

$2^{-3}$ = $\frac{1}{8}$

and so on and so forth.

another way to think of it is whenever an exponent is negative in a question, then you have to flip it with a fraction and the power that had the negative exponent now has a positive one, just on the other side of the fraction.

Example:

$3^{-4}$ = $\frac{1}{3^4}$

and with variable bases:

$x^{-y}$ = $\frac{1}{x^y}$

this rule can be easily applied to any power with a negative exponent

23. use the order of operation’s on expressions with powers involving negative exponents, and variable bases.

the answer would be $\frac{15}{x^6}$ because the do negatives add together due to the product law, and because of the negative law they must go

to the bottom part of the fraction but 15 doesn’t have a negative exponent, so it stays on top.

-Anything else I know about exponents

any power with a base of 1 equals 1.

2 thoughts on “Everything I Know About Exponents”

kaitlinw2016 says:

November 11, 2016 at 4:46 am

You did a great job on addressing all of the prescribed outcomes for exponents. Your explanations and examples were easy to understand, especially because you added lots of detail in each outcome. It was also made easy to understand you r explanations and examples when you used colours or bold to emphasize the important words. There was an error in 11 (use patterns to explain the zero law). Your explanation was that the numbers descend by 2. But 32 to 16 to 8 to 4 is not descending by 2 it is dividing by 2. Also you didn’t apply the negative exponent law to variable bases in 22, you only explained using integral bases. Other than that I didn’t find any other errors but in 23 I think it would’ve been better to evaluate your order of operations expression by solving and showing each step underneath the expression other than explaining in a paragraph. I thought that your examples in each outcome were well picked out for they really helped with your explanations. I also thought outcome 3 was swell done because you used a long detailed description to help understand the difference between the exponents and the base. Overall I thought your post was very good other than the couple errors.

thubbard says:

November 16, 2016 at 5:34 am

Hi Adam,
You still need a comment from a parent.

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Everything I Know About Exponents

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