1) How to represent repeated multiplication with exponents:  

(4)(4)(4)==64, (5)(5)==25

2) How powers represent repeated multiplication:

 =1*1=1, =3*3*3*3*3=243

3) The difference between the exponent and the base via building models of a given power:

                                                   

                              

4) The difference between two given powers are interchanged by using repeated multiplication:

=3*3=9

=2*2*2=8

5) Powers with integral bases (besides zero) and whole number exponents:  

=-8, =-81

6) The role of parentheses in powers by evaluating a given set of powers (e.g.) :

Anything in the brackets will go first, but with powers they act a little different;

=16, this means that -2  is getting multiplied by itself 4 times.

=-16, in this case -2 is still getting multiplied by itself by 4 times still but it will always be negative because since the base and exponent are together the coefficient will come into play (-1), so it would look like *-1*2*2*2*2=-16.

=-16, for this last scenario it is already all together so it would be the same (basically) as the scenario before this one.

7) The exponent laws for multiplying and dividing powers with the same base: 

multiplying: keep the base, add the exponents, multiply the coefficients  (e.g.)

 =, 2 * 3=6

dividing: keep the base, divide the coefficient, subtract the exponents

——————- =

8) The exponent laws for raising a product and quotient to an exponent:

When raising a product to an exponent, you must;

1. Multiply the exponents together (if there are any exponents in the brackets)
2. Multiply the base by itself _____ amount of times (depending on the exponent)
3. If there are variables with no exponents, you do the same as step 1

= (64 (16= 1024

When raising a quotient to an exponent, you just divide the exponents and subtract the exponents and keep the base.

=

9) The law for powers with an exponent of 0:

An exponent of 0 will always =1 but does not equal 1

10) Patterns to show that a power with an exponent of 0=1:

=4

divide by 2

=2

divide by 2

=1

divide by 2

=1/2

divide by 2

=1/4

11) The law for powers with negative exponents:

If you have a negative exponent, it will flip (become it’s reciprocal) (e.g.) =

12) Patterns to explain the negative exponent law:

In number 10 the pattern was every time the exponent went down by 1, we divided by the base. So that would mean that =1/3, =1/9

13) How to apply the exponent laws to powers with both integral and variable bases:

 =

14) How to identify the error in a simplification of an expression involving powers:

1. Know which law you are doing (product, power, quotient or zero)
2. Know the steps (do you add the exponents, subtract, multiply and do you multiply the coefficients or divide)

15) Use the order of operations on expressions with powers

   = =(16 

16) How to find the sum and difference of two powers:

Difference: –=16-8=8

Sum: +=49+125=174

17) How to identify the error in applying the order of operations in an incorrect solution:

1. Know the rule of BracketsExponentsDivisionMultiplicationAdditionSubtraction and what is stands for
2. Work out the problem to see if anything is incorrect

(e.g.) (3 (2= (12 (10= 120

The mistake here was that I multiplied the numerical base with the exponents

The correct equation would look like, (3 (2= (81 (32=2592

How to use powers to solve measurement problems:

(find the surface area), Answer:[(6)]= (16)(6)=96cm

(find the volume), Answer: =64cm

18) How to use powers to solve growth problems:

In a dog simulator, there are 4 puppies, they are so adorable that they triple their population every hour. How may will there be after 3 hours?

x=4=4(27)=108Puppies

Conclusion: 

I believe that is it; thank you for reading this essay.

 

XD