# Everything I know about exponents

Prescribed Learning Outcomes for Exponents:

1. Represent repeated multiplication with exponents:  The first step to represent a repeated multiplication with powers is to count how many time the number is multiply by it self. My example for this step is 17 x 17 x 17 x 17, you can count how many times 17 is times by it self, the answer is 4 times. The next step is now representing this with exponents, the way that I do this is taking the number that was timed by it self as the base and taking the number of how many times it was timed as the exponent. The example for this is 17 is the base and the 4 is the exponent = $17^4$
2. Describe how powers represent repeated multiplication: How I describe the way powers can be represent as repeated multiplication is very similar to learning outcome number 1. You start by identifying the base and the exponent of the power. The example I have; $7^5$, the base is 7 and the exponent is 5. With the information about the base and the exponent, you can start by taking the base and multiplying it by it’s self the amount of times the exponent said. My example is with your 7 and 5, the 7 being the base, you will times 7 x 7 x 7 x 7 x 7 because the exponent is 5.
3. Demonstrate the difference between the exponent and the base by building models of a given power:  For this learning outcome, I have two image to show how it works.         On the left is cube and let’s said that it is = to $7^3$ and the square is equaled to $7^2$. The number 7 is representing the side length of the cube and a side length of the square. The difference is my two examples is $7^3$ is that the answer is represent the volume of a cube. The $7^2$ is represent the area of a square.
4. Demonstrate the difference between two given powers in which the exponent and the base are interchanged by using repeated multiplication: Where the exponent and base are interchanged will not equal the same answer. The reason behind this is that you will always multiply the base by it’s self the amount of time of the exponent. So, with that $3^4$ will not be the same as $4^3$ because they have different repeated multiplication. $3^4$ = 3 x 3 x 3 x 3 (the answer being 81), when $4^3$ = 4 x 4 x 4 (the answer being 64). The answer is difference making the powers have a difference.
5. Evaluate powers with integral bases (excluding base 0) and whole numbers exponents:  You can evaluate powers with integral base and whole numbers exponents the same way. My examples are $-3^4$ and $5^8$. $-3^4$ =  -3 x -3 x -3 x -3 = 81 and $5^8$ = 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 = 390625. The both of those powers use the same steps, step one find the power and the exponent in the power. Next take the power and multiply it by it’s self the amount of times of the exponent. Then after doing the multiplication, you have your answer.
6. Explain the role of parentheses in powers by evaluating a given set of powers: The role of parentheses can change an answer to a power very drastically. My examples are $(-2)^4$ and $-2^4$, as you can see they look like almost the same power with one having parentheses and the other not having them. $(-2)^4$ in repeated multiplication is (-2) x (-2) x (-2) x (-2) = 16, when \$latex -2^4 in repeated multiplication form is -1 x -2 x -2 x -2 x -2 = -16. The answers are completely opposite.
7. Explain the exponent laws for multiplying and dividing powers with same base:  The exponent law for multiplying with the same base is to keep the base. Then add the exponents and multiply the coefficients. For example the equation; $2(x)^3$ x $3(x)^7$, you would first add 3 and 7 because they are the exponents, which equals 10. After that you are going to times the coefficients; 2 x 3 = 6. The answer will be $6(x)^(10)$. The exponent law for dividing powers of the same base is when you keep the base, subtract the exponents, and then divide the coefficients.  My example is $15(y)^4$ divide by $3(y)^2$, the first step would be to subtract the exponents; 4 – 2 = 2. Then you would divide the coefficients; 15 divide by 3 = 5. The answer to $15(y)^4$ divide by $3(y)^2$ = \$latex 3(y)^.
8. Explain the exponent laws for raising a product and quotient to an exponent: When you are raising a product, you are really doing a power to a power. For example $(8^2)^4$, the law is to keep the base the same and multiply the exponents. Your base is 8 so you keep it, and then multiply 2 x 4 which equals 8. So now you have $8^8$ which equals 16 777 216.
9. Explain the law for powers with an exponent of zero:  This law is pretty straight forward to explain, no matter what is your base if your exponent is 0 the answer will always be 1. My example for this law is $19^0$ and the answer is going to 1.
10. Use patterns to show that a power with an exponent of zero is equal to one: I am going to start with the base of 6 and an exponent of 9. $6^9$ = 10077696, $6^8$ = 1679616, $6^7$ = 279936, $6^6$ = 46656, $6^5$ = 7776, $6^4$ = 1296, $6^3$ = 216, $6^2$ = 36, $6^1$ = 6, $6^0$ = 1. The way that we can use the pattern to explain the zero law. If you divided the answer of $6^9$ by 6 it will be the answer of the next $6^8$. 10077696 divided by 6 equals 167916, if you keep doing that until $6^1$  = 6, six divided by 6 equals 1. That is the reason any base to the exponent of 0 equals 1.
11. Explain the law for powers with negative exponents:  The law for is to find the reciprocal of the base of the power, then keep the exponent with the original base, and next drop the negative sign. My example is $4^-3$. The way to find the reciprocal is first to put the power in a fraction; $4^-3/1$. Then the reciprocal will be $1/4^3$, however now you drop the negative sign. The answer to my example is 1/64.
12. Use patterns to explain the negative exponent law: For showing how to explain the negative exponent law I am going to start of with the power $3^3$, we know this equals 27. The next one would be $3^2$ which equals 9. After $3^1$ equals 3, and then $3^0$ equals 1. With the start of the patterns we can already figure out how to do the next bit of the pattern. You can take the 27 and divide it by 3, which equals 9, the next answer of the pattern. If you continue doing that, 9 divide by 3 equals 3, 3 divide by 3 equals 1. So now, if you take 1 and divide it by 3 it equals 1/3, which is also the answer to $3^-1$. Next 1/3 divide by 3 equals 1/9, the answer of $3^-2$.
13. I can apply the exponent laws to powers with both integral and variable bases: First of all an integral base is a base that could be any number, such as …-2, -1, 0, 1, 2, 3… A variable base is a base that is represented by a letter because the value of the base is unknown. My example for the integral base is $(-4)^2$ x $(-4)^3$. The first step is to use the product law, add the exponents, 2 + 3 = 5, keep the base. Now the power is $(-4)^5$, which equals -1024. The example I have for variable bases is $y^-2$ divide by $y^-4$. First keep the base, then subtract the exponents; -2 – (-4) = 2. Now the answer is $y^2$.
14. I can identify the error in a simplification of an expression involving powers:  In this expression I can identify the error in simplification; $3^5$ x $3^3$ = $3^7$. The way I can find the error is by following the product law. The first step is to keep the base; 3. Then you add the exponents; 5 + 3 = 8. The answer is $3^8$.
15. Use the order of operations on expressions with powers: When using the order of operations, follow the acronym; BEDMAS: brackets, exponents, division, multiplication, addition, subtraction. My example is $2^5$ – 7 x  $(-5 + 2)^3$. The first thing you need to do is the adding in the brackets; -5 + 2 = -3. Next, you are going to do the exponents; $(-3)^3$ = -27, and $2^5$ = 32. After that you are going to do the subtraction; 32 – (-27) = 59. The answer to $2^5$ – 7 x $(-5 + 2)^3$ = 59.
16. Determine the sum and difference of two power:  Let start with the sum of two powers first. My example is $2^4$ + $3^5$, the first step is to solve each power. $2^4$ = 16 and $3^5$ = 243. The step is to add the answers of the powers. 16 + 243 = 259. When finding the difference of two powers, you have very similar step to finding the sum. The example I have is $2^9$$3^2$. Your first step again is to find each answer to the powers. $2^9$ = 512 and $3^2$ = 9. Now, subtract the two answers. 512 – 9 = 203.
17. Identify the error in applying the order of operations in an incorrect solution:  The example I have for the incorrect solution is $(-2 + 4)^6$ – 5 x $2^3$, and first you did the brackets; (-2 + 4) = 2. Then the exponents; $2^6$ =  64 and $2^3$ = 8. Next you did the subtraction; 64 – 5 = 59. Then finally you did the multiplication; 59 x 8 = 472. Now I am going to find out when the error is in the operation. The first you need to do is to follow BEDMAS. I know by looking at the operation, I need to do brackets, then exponents, then multiplication, and finally subtraction. Already I can see the error, when I subtract was when I was supposed to multiply. If I was to do it right 5 x 8 = 40. Then 64 – 40 = 24. The correct answer to the order of operations is 24.
18. Use powers to solve problems (measurement problems):  My example is; if you had a 7cm side length of a shaded squared with a square that isn’t shaded on the inside that has a side length of 3cm, much of the shaded part is left. This question written out is $7^2cm$$3^2cm$, and that would equal 49cm – 9cm = $20cm^2$.
19. Use powers to solve problems (growth problems):  The example for this learning outcome is a factory triples the number of toys each hour. There is 25 toys so how many toys will the factory make after five hours? 1 hour = 25 x 3 = 75. 2 hours = 25 x 3 x 3 or $25 x 3^2$ = 225. 3 hours = 25 x 3 x 3 x 3 or $25 x 3^3$ = 675. 4 hours = 25 x 3 x 3 x 3 x 3 or $25 x 3^4$ = 2025. 5 hours = 25 x 3 x 3 x 3 x 3 x 3 or \$latex 25 x 3^5 = 6075.
20. Applying the order of operations with powers involving negative exponents and variable bases:  My example is $(-10)^-x$ x (y). The first step is to do change it into $y(-10)^-x$. Then put it into the reciprocal of the base = $1/y(-10)^x$