This unit we have learned about exponents so in this blog I will be answering questions and explaining some things we’ve learned.

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1.Represent repeated multiplication with exponents

Powers are basically a shortcut for repeated multiplication. Instead of writing 3*3*3 you can write 3 to the exponent of 3 because that is how many times you are multiplying it. This is a much easier way to write in equations.

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3. Demonstrate the difference between the exponent and the base by building models of a given power, such as two to the power of three and three to the power of two.

The base equals the base number and the exponent represents how many times you multiply the base. Two to the power of three is definitely not the same as three to the exponent of two because the first one equals 2 times 2 times 2 and the second one equals 3 times 3. 2 to the power of three equals to the volume of a cube with a side length of 2. 3 to the exponent of two is the same as the area of a square with the side length of 3. 

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

Basically, this rule is simple. When you have a negative base in brackets and the exponent is positive, the answer is positive. When you have a negative base in brackets with a odd exponent, the answer is negative and when you have a positive base, it doesn’t matter if the exponent is even or odd because the answer is always positive. 

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

The rule for multiplying and diving powers with the same base is easy. For multiplication, you simply add the two exponents together to get your answer. For division, you have to subtract the exponents to get your answer. You always keep the base. I have drawn examples to give a visual explanation.

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

I made some examples of situations where the exponent zero could be used. Any number to the exponent zero equals one unless the base is zero. Even if there is a negative sign, as long it is in the brackets, the answer is one. Now the only time that it would equal something other than one is if there is a negative sign outside of the brackets,  then the answer is negative one. For my other examples I used the product and quotient rule to create two examples.

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11. Explain the law for powers with negative exponents. 

Basically, when you have a negative exponent in order to get rid of it you need to place it on the other side of the fraction. Ex. If it is in the denominator you put it in the numerator and if it is in the numerator you put it I the denominator. If it isn’t a fraction write the number over one and then you switch the fraction so it is one over the number as shown in some of my examples. 

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

Multiplying and dividing exponents with the same base is easy and simple. For multiplying exponents you have to keep the base and add the exponents together like my example above. For dividing exponents you again keep the base and this time you subtract the exponents. I also solved a more complicated equation but pretty much using the same rules. Like I did in number seven, I added the exponents for the same base, and I multiplied the coefficients. Then, I simplified it by eliminating x since it had the same exponent in the numerator and denominator the simplified x would be x to the exponent 0 and we already that equals to one and in an equation where x is multiplied multiplying it by one wouldn’t change the number so it can be eliminated. The rule for exponents in fractions is that if the same base is in the numerator and denominator, you subtract the exponent in the numerator from the exponent in the denominator and that is what I did for y. Since the answer to this was a negative exponent, like I said in my example before you put y on the opposite part of the fraction in order to make the exponent positive. So my answer was 48 over y to the exponent of 2.

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

This time I made up an equation to solve using order of operations. I made sure to follow BEDMAS. brackets are first, but I couldn’t calculate before dealing with the exponents so I just made sure the brackets stayed in the right place. Then, I calculated the numbers in the brackets which equaled 1 because 5-4 equals 1. Then I had to calculate the exponents so after that it equaled 6*16*1 which is equivalent to 6*16 so the answer is 96.

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

This time I calculated the same question, but I didn’t use order of operations. I calculated the exponents but completely ignored the brackets and did the exponents first only keeping the brackets in place. I ended up getting 6*16*9 instead of the correct way from my previous question where I evaluated to I side of the brackets first which equaled to 5-4 and then I got 1 and had 1 to the exponent of 2 which equaled 1 and not 9. So instead of having my answer be 864 it should be 96.

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19. Use powers to solve problems (growth problems)

for this word problem example I had to use the starting amount of rain, by how much the amount gets bigger every hour and the amount of hours I have to calculate. To determine how much more rain there will be in four hours, I had to do three to the power of four. (the base is three because it triples every hour). Then, I multiplied that by four because the amount of rain there was in the beginning was four. That all equaled to 324 ml/h. I know realistically there wouldn’t be 324 milliliters of rain, but it is the answer if the rain were to keep tripling every hour.

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some other things I know…

Another thing I would like to explain is fractions with negative exponents. If a fraction has a negative exponent applied to the whole thing, when you flip the fraction the exponent becomes positive.

Something else that I would like to mention is that when two numbers from a fraction from the numerator and denominator both have the same base and negative exponents, when you flip the two numbers the fraction will end up having positive exponents. Then, you subtract the exponents if there is the same base to get your answer. In this case the answer was x.

Also, I know that when there is an example such as (2^2 * 2^2)^2 you add the exponents on the inside of the brackets for multiplication and that equaled to (2^4)^2 and since I also need to calculate the exponent on the outside I multiplied the powers: 4*2 and that equaled 8 so now it’s 2^8. I calculated this and the answer is 256.

In conclusion, during this unit I’ve pretty much understood everything about exponents that we’ve learned in class and with a little more practice I think that I will be even better at getting accurate answers.

Here is a link to my partner’s blog:

Sophia’s blog

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