Everything I Know About Exponents Math 9 H 2016

1.Represent repeated multiplication with exponents.

The repeating factor of this multiplication equation would be the base of the power while the number of times we multiply it by would be the exponent.

Ex. $3\cdot3\cdot3\cdot3=3^4$

2.Describe how powers represent repeated multiplication.

The base of the power determines the number we multiple by while the exponent shows us the number of times we multiply it by itself. Make sure that you never apply this rule to powers with negative exponents since it will not provide you with the right answer. Another common mistake is that people believe that $5^5=5\cdot5=25$ while the correct answer is $5^5=5\cdot5\cdot5\cdot5\cdot5=3125$ .

Ex. $4^4=4\cdot4\cdot4\cdot4$

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 first model represents the equation $3\cdot3=3^2$ while the second one shows us $2\cdot2\cdot2=2^3$ . Although they may seem visually similar they display two various mathematical formulas. It is important not to confuse the two because the answers are completely different.

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 $3^2$ .

(Ex. $2\cdot2\cdot2=2^3$ )

(Ex. $3\cdot3=3^2$ )

The difference between these two equations is that $2^3$ tells us that we have to multiply two by itself three times, giving us an answer of eight while $3^2$ states that we have to multiply three by itself two times giving us an answer of nine.

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

Here are some examples of powers with integral bases and whole number exponents.

Ex. $6^4=1296$

Ex. $(-3)^3=-27$

Ex. $-4^4=-256$

Ex. $0^6=undefined$

Ex. $1^{12}=1$

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

$-2^4=(-1)\cdot2\cdot2\cdot2\cdot2=-16$

$(-2^4)=(-1)\cdot(2)\cdot(2)\cdot(2)\cdot(2)=-16$

$(-2)^4=(-2)\cdot(-2)\cdot(-2)\cdot(-2)=16$

When you are asked to simplify these equations you have to be very cautious about where the parentheses are placed because they can give multiple different answers. In the first question since there is no brackets the base is actually 2 and that is what gets repeated while the $-$ is represented as a -1. The second equation is very similar to the first one since the parentheses are outside the the exponent. In this case we can just drop the brackets and proceed the same way we would as with the first question. For the last equation since the parentheses are surrounding -2 that is what gets repeated and since we know that when we multiply by and even number of negative exponents the negatives cancels each other out and our answer will be positive

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

The Product Law: The law says that you first have to keep your base, then add together the exponents and finally multiply your coefficients. Keep in mind that it only works with multiplying powers with the same base.

Ex. $5^2\cdot5^4=5^{2+4}=5^6=15625$

Ex.2. $3(4^5)\cdot2(4^2)=3\cdot2(4^{5+2})=6(4^7)=6(16384)=98304$

Quotient Law: The law states that you first have to keep your base, then subtract the exponents and finally divide the coefficients. Keep in mind that it only works with dividing powers with the same base.

Ex. $3^{13}\div3^{7}=3^{13-7}=3^6=729$

Ex.2. $12(4^{14})\div3(4^9)=12\div3(4^{14-9})=4(4^5)=4(1024)=4096$

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

When you are using the exponent law to raise a product or a quotient to an exponent you have to keep the base and multiply the exponents. You then take the coefficients and individually raise them to the exponent. When solving this for a quotient make sure that you do it for both the numerator and the denominator.

$(3x^2y^4)^3=(3^3x^{2\cdot3}y^{4\cdot 3})=(27x^6y^{12})$

$(\frac{1x^3}{3y^4}) ^2=(\frac{1x^6}{9y^8})$

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

When you are evaluating powers with an exponent of zero (Ex. $6^0$ ) where the base doesn’t equal to zero the answer will always be one. One common mistake is that people believe that $7^0=0$ since $7\cdot0=0$ however that is not true.

Ex. $7^0=1$

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

There are actually several ways to explain the the Zero Law. The first one is by using a pattern such $5^4=625$ , $5^3=125$ , $5^2=25$ , $5^1=5$ , $5^0=1$ . This pattern shows us that each time the exponent decreases the previous answer answer is divided by the base. Since $5^1=5$ we would divide the 5 by itself and get the answer of 1. Another way to explain this would be using BEDMAS ( $5^5\div5^5= 3125\div3125=1$ ) and also the Quotient Law ( $5^5\div5^5=5^{5-5}=5^0=1$ ) . Since both $5^0$ and 1 are the answers to the same question they must be equal, therefore $5^0=1$ .

11. Explain the law for powers with negative exponents.

When you are solving an equation with a negative exponent such as $4^{-3}$ you are required to reciprocal the power and change the exponent to a positive. This law only applies to bases that do not equal to zero.

Ex. $4^{-3}=\frac{1}{4^3}=\frac{1}{64}$

Ex. $\frac{1}{4^{-3}}=4^3=64$

12. Use patterns to explain the negative exponent law.

As you can see in the pattern below it demonstrates the connection between positive and negative integers. As I mentioned in the previous patterns, each time the exponent decreases the previous answer is divided by the base. Same goes with negative exponents.

$3^2=9$ , $3^1=3$ , $3^0=1$ , $3^{-1}=\frac{1}{3^1}=\frac{1}{3}$ , $3^{-2}=\frac{1}{3^2}=\frac{1}{9}$

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

Here are just a few of the many examples.

Ex. $\frac{(4^2)^4(3^6)^3}{(4^3)^2(3^2)^4}=\frac{(4^8)(3^{18})}{(4^6)(3^8)}=4^{8-6}\cdot3^{18-8}=4^2\cdot3^{10}=8\cdot59049=944784$

Ex. $7^{3}\cdot7^{-7}=7^{3+-7}=7^{-4}=\frac{1}{7^4}=\frac{1}{2401}$

Ex. $(-2)^3(4)^0(-2)^5(4)=(-2)^3+5\cdot4=256\cdot4=1024$

Ex. $(x^4)^2\cdot(x^3)=(x^8)\cdot(x^3)=x^{8+3}=x^{11}$

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

$[5(x^2y^4)]^2\cdot[7(x^4y^6)]^3=[5\cdot2(x^{2\cdot2}y^{4\cdot2})]\cdot[7\cdot3(x^{4\cdot3}y^{6\cdot3})]=[10(x^4y^8)]\cdot[21(x^{12}y^{18})]=[10\cdot21(x^{4+12}y^{8+18})]=[210(x^{16}y^{26})]$

When this equation was simplified one very important mistake had been made. Instead of raising the coefficient to the exponent they multiplied it giving them a totally different answer. When completing this question be very careful and make sure that you are using the power law.

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

Ex. $-2(-15-4^2)+4(2+3)^3=-2(-15-16)+4(5)^3=-2(-31)+4(125)=62+500=562$

Ex.2. $4^2-8\div2+(-3^2)=16-8\div2+[-1(3)(3)]=16-4-9=3$

When you are completing these types of equations be very careful and follow all the Exponent Laws and BEDMAS.

16. Determine the sum and difference of two powers.

When you are trying to determine the sum or difference of two powers, just use BEDMAS.

Ex. $3^4+5^2=81+25=106$

Ex. $2^2+6^3=4+216=220$

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

Ex. $(8+16\div4)+2^3-5=$ $(24\div4)$ $+2^3-5=6+8-5=14-5=9$

In this equation you can observe that we first added in the brackets and then divided however we were required to the division before when applying the Order of Operations to our questions.

18. Use powers to solve problems (measurement problems).

In this question you are required to find to the the area of the shaded region. You first have to find the area of the larger square and the area of the smaller square inside it, once you find the areas you subtract them to find the area of the shaded region.

$A=3^2-2^2=6-4=2cm^2$

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 and what is our beginning number.In the example below we are starting of with 30 bacteria that double each hour. You would first start off with your total number of bacteria in the beginning and multiply it by by how many times it grow and the exponent would represent the amount of time that has passed.

Ex. 2 hours = $30\cdot2^2=120$

Ex. 5 hours = $30\cdot2^5=960$

Ex.n hours= $30\cdot2^n$

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

Here is one of the examples:

$\frac{(4)^{-2}x^2c^{-3}}{5^{-2}b^{-2}}=\frac{x^25^2b^2}{4^2c^3}=\frac{25b^2x^2}{16c^3}$

In this equation I used laws such as the Product Law, Quotient Law and BEDMAS. When completing this question make sure that you do everything in order and by careful when operation with negative exponents.

21. Anything else that you know about exponents.

Something else I know about exponents is that one raised to any exponent is always going to be one.

Ex. $1^5=1$ , $1^{-5}=1$ , $1^0=1$

Another things is that anything to the power of one always equals the base.

Ex. $6^1=6$ , $10^1=10$

3 comments

jaynab2016 says:

November 11, 2016 at 5:16 pm

Hi Evelina, Over all your blog was great. I liked all the time and thought that you have put into this, and leaving good examples and explanations in each question, and addressed all the learning outcomes. You do have a few simple mistakes in some of your work. In number 7, in your second example I think you may have made a simple mistake. At the last stage, you had $6(4^7)$ first with the answer of 6(616348). Your final answer is correct, however I think you may have accidentally added the 6 at the beginning of the answer before, because it doesn’t make sense that $6\times616348$ equals to 98304. Another thing that I might add, is the very last question, “Anything else that I know about exponents.” Even though this may not be a actual question, you may want to consider putting one or two things there, like maybe “any base with an exponent of one, is always equal to the base”. Besides this, I think that You did a wonderful job completing this assignment, and everything is presented and explained very clear. Great job Evelina

Svetlana says:

November 15, 2016 at 3:44 pm

I can see how much time and effort you have put into this post while trying to explain all the learning outcomes. You also demonstrated the evidence of you learning and progressing throughout the course of this unit. I am very proud to see your latest achievements, keep up the good work.

thubbard says:

November 16, 2016 at 1:59 am

Thanks for making the changes mentioned by Jayna, check your Onenote for more specific feedback.

Evelina's Blog

Everything I Know About Exponents Math 9 H 2016

3 comments

Leave a Reply Cancel reply