# Everything I Know About Exponents!!!

#1. Represent repeated Multiplication with exponents
To represent repeated multiplication with exponents you start by counting the amount of times your base repeats; 2x2x2 = to 3 2’s
You then make the 3 the exponent and the 2 the base; $2^3$

#2. Describe how powers represent repeated multiplication
Well, when demonstrating how to write repeated multiplication you show the amount of times the base repeats; 2x2x2x2 But in a power you shorten the process by writing 2x2x2x2 in a shorter fashion; $2^4$ 2=base (# you multiply) 4=exponent (# times your base repeats).

#3. Demonstrate the difference between the exponent and the base by building models of a given power, such as $2^3$ and $2^2$
The model for a base to the exponent of three is cubed, So the model will have 6 sides that each equal $base^2$ (a) For (a) each side is equal to 4
a)
b)
For $2^2$ its real meaning is the area of the square, 2 x 2=4 so the area is $4u^2$ (b) and it will show the area of that one surface.

#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 $3^2$
$2^3$ = 8 while $3^2$ = 9
They would not be interchangeable because it is not 2×3 it is 2x2x2 and it’s not 3×2 it is 3×3. We multiply the base by itself, the number of times the exponent is.

#5. Evaluate powers with integral bases (excluding base 0) and whole number exponents
To evaluate integral bases and whole numbers you start by multiplying the base by itself as many times as the exponents say you need too; 5 x 5 x 5 x 5 = 625 which is equal to $5^4$. So you multiply 5 by itself 4 times.

#6. Explain the role of parentheses in powers by evaluating a given set of powers such as $(-2)^4$, $(-2^4)$ and $-2^4$
For a set of powers the brackets indicate a lot in these examples;

$(-3)^2$ represents (-3)x(-3)= 9
$(-3^2)$ represents (-1)x3x3= -9
$-3^2$ represents (-1)x3x3 = -9
The parentheses really indicate whether your doing (-1) multiplied by your $base^e$ or the literal base ex: (-3)x(-3) / $(-3)^2$

#7. Explain the exponent laws for multiplying and dividing powers with the same base
When you multiply a power, your exponent law the rule is 1; Keep the base 2; Add the exponents 3; multiply the coefficients. So what that means is if you have $5^2$ x $5^3$ you keep 5 as the base and add the exponents. $5^3+^2$ = $5^5$ = 3,125
When you divide a power, your exponent law rule is 1; Keep the base 2; subtract the exponents 3; divide the coefficients. First you would take $5^3$ divided into $5^2$ and, keeping the base, subtract the exponents; $5^3-^2$ = $5^1$ = 5

#8. Explain the exponent laws for raising a product and quotient to an exponent
When raising an a product or quotient to an exponent you have a few rules. You keep the base if they have the same base, multiply the exponents and then apply the outside exponent to the coefficient.
Ex: (2 $2^2)^2$ divided by (4 $2^3)^2$

First you would square the exponent and multiply the two powers; $2^2$ $2^2x^2$. Then you would get 2 x 2 = 4 and $2^4$ = 16 you would then do 4 x 16 = 64 (Coefficient x the power)

Next you would do the second part of the question using the same steps; $4^2$ $2^3x^2$. This would give you 4 x 4 = 16 and $2^6$ = 64 so 16 x 64 = 1024

Finally you would do 1024 divided by 64 = 16

#9. Explain the law for powers with an exponent of zero
Any power with the exponent of 0 automatically equals 1; $123,456^0$ = 1                          Except when its $0^0$ = 0

#10. Use patterns to show that a power with an exponent of zero is equal to one
You can use the quotient law (subtracting the exponents) and also using BEDMAS works.
Ex: ( $2^2$ )divided into ( $2^2$ ) = 4 divided into 4 = 1
( $2^2$ ) divided into ( $2^2$ ) = $2^2-^2$ = $2^0$ = 1

#11. Explain the law for powers with negative exponents
When you have a power with a negative exponent you have to flip the power into a fraction (the power as the denominator). This turns the power into a positive.
Ex: $2-^3$ = 1/ $3^2$ = 1/9

#12. Use patterns to explain the negative exponent law
$3^3$ = 27
$3^2$ = 9
$3^1$ = 3
$3^0$ = 1
$3-^1$ = 1/3
$3-^2$ = 1/9
$3-^3$ = 1/27
The pattern will continue forever in both directions dividing by 3.

#13. I can apply the exponent laws to powers with both integral and variable bases
What this means is that I can do questions with integers (whole numbers) and variable (changing bases; exponents/powers) bases. To solve problems like these you use all the exponent laws including; quotient, product, power, zero law and negative exponents law.

#14. I can identify the error in a simplification of an expression involving powers
I can easily see errors that may occur in simplification of expressions. Using the laws previously mentioned I can define and detect problems in a resolution.
For example if you have $5^3$ x $5^2$ and someone put their steps out like so; $5^3$ x $5^2$=125 x 25 = 3,125

An incorrect equation would be if you used the wrong law; $3^6$ x $3^3$ = $3^6-^3$ = $3^3$ = 27

The correct way would  be to use the product law (adding the exponents) not the quotient law (subtracting the exponents).

#15. Use the order of operations on expressions with powers
When subtracting and adding you should 100% use BEDMAS but if you have question where you can take short cuts like a quotient, product or power law that is the best way to do it.
Ex: $2^4$ + $5^3$ = 16 + 125 = 141
Ex: $5^3$ divided into $5^2$; $5^3-^2$ = $5^1$ = 5

#16. Determine the sum and difference of two powers
When finding the sum or difference of two powers you always use BEDMAS. After you would de the brackets you start with the exponent by (if the bases is the same) adding the two exponents together to make a single power. Then Evaluate.
Ex: $5^2$ + $5^4$ , 25 + 625 = 650
Ex: $2^2$ + $2^5$ , 4 + 32 = 36

#17. Identify the error in applying the order of operations in an incorrect solution
Always be sure to follow the rules of BEDMAS…
$5^2$ – 6 + 12 = 18 – 25 = -7 that is incorrect because you have to do the exponent #1st in this equation and you need to do all operations from left to right. The answer was 19 while it should have been 31; 25 – 6 = 19 + 12 = 31

#18. Use powers to solve problems (measurement problems)
Many ways you can use powers for measurements/measurement problems are; area (squared) and volume (cubed). When trying to find the area you find one side length and “square” it so it is showing the area of your square, this is also the same as length x width. When using powers to find the volume you find the “cubed” which is the same as length x width x height.
Area: side length = 2 which is equal to $2^2$ = 4
Volume: length = 3 width = 3 height = 3 which is equal to $3^3$ = 27

#19. Use powers to solve problems (growth problems)
When solving growth problems you use powers to double,triple the base (product). You need to know what you start with, how much it grows by and how long it grows.
Take this example; you have 2 apples and every hour they triple. How many apples do you have after 5 hours?
5th hour: 2 x $3^5$ = 243 x 2 = 486
The formula is your coefficient (apples) multiplied by your triple/double to the exponent (hour).

#20. Applying the order of operations on expressions with powers involving negative exponents and variable bases
The law for negative exponents is you flip the power into its reciprocal fraction to make a positive exponent.
Ex: $2-^1$ + $2^3$$6^2$
= 1/ $2^1$ + 8/1 – 36/1
= 1/2 + 8/1 – 36/1
= 1/2 + 16/2 – 72/2
= 17/2 – 72/2
= -55/2