Everything I Know about Exponents

katie-yewonc2016Math 9

Prescribed Learning Outcomes for Exponents:

1) Represent repeated multiplication with exponents

You count the number of times you multiply the number by itself which becomes the exponent and the number itself becomes the base.

Example: 5\cdot5\cdot5 = 5^3

2) Describe how powers represent repeated multiplication

The simplified version of repeated multiplication (power) is spread out to show how many bases there are. The little number at the top of the power( exponent) represents how many of the bases there is.

Example: 2^4= 2\cdot2\cdot2\cdot2

3) Demonstrate the difference between the exponent and the base by building models of a given power, such as 3^2 and 2^3 .

If two powers have the same numbers and are just switched around, that doesn’t mean they have the same answer. 3^2 has an exponent of 2 (squared) so you would know the model has to be a 2D, square. 2^3 has an exponent of 3 (cubed) so you would know the model has to be a 3D, cube.

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4) Demonstrate the difference between two given powers in which the exponent and the base are interchanged by using repeated multiplication, such as 3^2 and 2^3.

The exponents and bases are switched but just because they are the same numbers, it doesn’t always mean they equal the same thing. The repeated multiplication for the 2 comparisons also are different.

Example: 2^4= 2\cdot2\cdot2\cdot2= 16   4^2= 4\cdot4= 16

1^2= 1\cdot1= 1    2^1= 2

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

We have to find the answer for the power and you have to remember to multiply the number of bases, not add since it is exponents.

Example: 5^2= 5\cdot5= 25

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

In (-2)^4, the exponent is outside of the bracket so it means -2 is multiplied by itself 4 times which becomes a +16. In (-2^4), the exponents are inside the brackets with the base so it is basically saying it is (-1 times 2^4) which becomes a -16. In -2^4, there are no brackets so it means -1 times 2^4. Overall, the brackets make the bases multiply with the negatives, without it, or just one.

Example: (-3)^2=(-3)\cdot(-3)=+9   (-3^2)= (-1\cdot3\cdot3)= -9       -3^2= -1\cdot3\cdot3

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

Multiplying: If it is the same base, you can just leave the base as it is and ADD the exponents together. (Product Law)

Dividing: You leave the base as it is if it’s the same and SUBTRACT the exponents. (Quotient Law)

Example: Multiplying- 2^3\cdot2^2= 2^5

Dividing- 3^6\div3^4= 3^2

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

Raising a product and quotient to an exponent is called the power law. You would have to keep the base then multiply the exponents together. If there was a coefficient then you would also do that to the power of the exponent too.

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

Answers for question with an exponent of zero is always 1 because if a base to the power of 1 is the base, then a 0 exponent is 1 since base divided by base = 1.

Example: Since 4^1= 4, 4^0 is 4\div4= 1

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

If you look at every power with the same base, (whichever direction you look from) it multiplies/divides by the base which eventually meets the exponent of zero which becomes one. Base divided by the base equals one.

Example: Since 6^1= 6, 6^0 is 6\div6= 1

11) Explain the law for powers with negative exponents.

If it is a negative exponent, you do the reciprocal of the base and the exponent becomes a positive. If the base isn’t a fraction, the exponent can be below in the denominator spot with the base without using brackets  and then writing the exponent. When the base is a fraction you would write the reciprocal inside brackets and outside them, you would write the exponent to show it’s both for the numerator and the denominator.

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12) Use patterns to explain the negative exponent law.

A pattern that can be helpful is to think that 2^3= 8, 2^2=4, 2^1= 2, 2^0=1. The pattern here is that every time one exponent decreases, the answer divides by the base which is 2. If 2^0=1 then dividing 1 by 2 will equal \frac{1}{2} which is for 2^-1. Basically the pattern is that you keep dividing by the base every time one exponent decreases and you do it until it reaches the fractions.

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

The same exponent laws to powers with variable bases are applied the same way as the integral bases. Product Law: Multiplication question, 1) if same base, keep the base, 2) add the exponents. Quotient Law: Division question, 1) if same base, keep the base, 2)subtract the exponents. Power Law: If a product or quotient is raised to an exponent, multiply the exponents.

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

I can identify errors in a simplification of an expression involving powers because I know all and fully understand the exponent laws/the steps for each one. Once you know the steps and know how to identify each law then you can solve power questions without getting confused.

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

Order of operations is basically using BEDMAS (brackets, exponents, division, multiplication, addition, subtraction) in the order that it appears in (multiplying/dividing & adding/subtracting: order doesn’t matter). You always have to look carefully at the question which step you have to do first.

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16) Determine the sum and difference of two powers.

When finding the sum and difference of 2 powers, you just apply BEDMAS to find the answers. There is no other quicker way to do it.

Example: 1) 2^3+ 5^2= 8+25= 33

2) 6^2– 3^3= 36-27= 9

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

When trying to identify the error in the order of operations, a mistake people can make is forgetting to do BEDMAS and just evaluating the problem from left to right. It could give a totally different answer which is a common mistake people make.

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18) Use powers to solve problems (measurement problems)

Powers can be used in simple problems like measurement problems. A measurement problem could be trying to find the shaded area of a bigger square when a smaller square is covering a part of it. Since squares have equal length sides, you could make the equation simpler by using exponents too. If it was trying to find the volume, you would have to use the exponent: ^3 which is cubed for volume. If you were trying to find the other side of the triangle, you would use the Pythagorean Theorem which involves exponents.

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 19) Use powers to solve problems (growth problems) 

When a problem says something grows in a repeated pattern, for example, “triples” then the base would be 3. Whichever word they say becomes the base (doubles, quadruples, etc.) and the exponent would be the time (every hours,minute, seconds, etc.)

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

For every question you always have to remember that you have to follow the BEDMAS rules.You should do the coefficients, then do any of the exponent laws excluding the negative exponent law and then you do it. You have to do the other exponent laws first before you do the negative exponent law.

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Others things I know:

If the any of the exponent laws have a coefficient, they need to be evaluated separately from the bases and exponents

Coefficients are a simpler way of saying repeated addition.

5 thoughts on “Everything I Know about Exponents”

  1. Hi Katie! Your blog post looks amazing! It is very nicely organized and detailed. Some things to keep in mind are try and make the sentences even simpler and easier to understand ex.#9 I had some difficulty understanding it. Another thing for #10 and other things I know, could you show the patterns/steps more clearly. But overall excellent post! Great work!

    Reply

    1. Hi Ms Hubbard. My dad commented on it by the due date it was supposed to be done by, but I realized it wouldn’t show up on other people’s accounts since he wrote on my own account. It is fixed now.

      Reply

  3. Katie understands exponents pretty well. She knows how the each laws of the exponents works as shown on her own examples.Katie realized my comment did not show in public since I did it on her own account although I commented by the due date.
    Very good Katie!

    Reply

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