Everything I know about exponents

1) Represent repeated multiplication with exponents

To represent repeated multiplication in an exponential form, for example: 4x4x4. There are three 4s in this multiplication. So to write it as an exponent , you write down the base which is 4, and you write 3 as the exponent since that’s how many 4s there are. So it is $4^3$ .

2) Describe how powers represent repeated multiplication

Power: an expression made up of a base and an exponent.

A base and an exponent make a power, ex: $4^2$ , this example also represents 4×4. because base is the number you multiply by itself, and the exponent is the number of times you multiply the base.

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

For $2^3$ , represent it to 2x2x2, since 2 is the number you multiply, the BASE. and 3 times because thats the EXPONENT, the number of times you multiply.

For $3^2$ ,it is represented into 3×3, since 3 is the BASE, the number you multiply, and 2 times because that’s the EXPONENT.

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 $3^2$ .

$2^3$ means 2x2x2, three 2s. and $3^2$ means 3×3, two 3s.

Also, $3^4$ is different to 3×4, $3^4$ is 3x3x3x3, multiply by the number of times the exponent is. And 3×4 means just 3×4.

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

evaluate: $(-2)^3$ It means (-2)(-2)(-2), so three -2s.

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 $(-2)^4$ , the -2 is in brackets which means it stays together, so it represents (-2)(-2)(-2)(-2), so it equals 16. For $(-2^4)$ , the $(-2^4)$ is in brackets, and in that case the negative in the front becomes a COEFFICIENT, the negative sign basically means -1 in this power. So you would do -1x2x2x2x2, which would be -16. As you see, the answers are different even though it’s just the brackets that changed. For $-2^4$ , It is the same except this power doesn’t have a bracket which doesn’t make a lot of difference. So it would be -1x2x2x2x2, which is -16.

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

Product law: 1. Keep the BASE. 2. Add the exponents. 3. Multiply coefficients.

ex: $4^2$ x $4^3$ , keep the base so the base is 4, add the exponents: 2+3=5. the exponent is 5. so the answer is $4^5$ .

Quotient law: 1. Keep the base. 2. subtract the exponents . 3. divide coefficients.

ex: $3^4$ / $3^2$ keep the 3 because it is the base and subtract the exponents, 4-2=2. so the answer is $3^2$ . If you think about it in repeated multiplication: 3x3x3x3 would be the numerator and the denominator would be 3×3. Then you can crossed out two of the 3s on top and bottom because they cross each other out. so the answer would be $3^2$ .

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

Power law says: 1.Keep the base, 2.multiply the exponents 3. don’t forget the coefficient.

ex: $(2^3)^3$ Keep the 2 since its the base, multiply the exponents, 3×3=9. so the answer to this would be $2^9$ . If you think about it as repeated multiplication, it would be (2x2x2)(2x2x2)(2x2x2). so if you count all of the 2, it will be $2^9$ .

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

The zero exponent law says :a power with an exponent of 0 equal 1. If you use bedmas to solve $2^3$ / $2^3$ , it is 8/8 =1. And if you use quotient law to solve it, $2^{3-3}$ = $2^0$ . Since $2^0$ and 1 is the answer to the same question, it means they must be equal. so $2^0$ =1.

ex: $5^3$ / $5^3$ , keep 5 since it’s the base and subtract exponents, 3-3=0, so $5^0$ which equals 1. Or use Bedmas: 15/15 =1.

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

$2^5$ =32 $2^4$ =16 $2^3$ =8 $2^2$ =4 $2^1$ =2 $2^0$ =1

$-3^5$ =-235 $-3^4$ =81 $-3^3$ =-27 $-3^2$ =9 $-3^1$ =-3 $-3^0$ =1. The pattern is that as the exponent decreases by 1, the answer is divided by the base.

11) Explain the law for powers with negative exponents.

rules for negative exponents: 1.reciprocal base 2.make exponent positive.

ex: $2^{-3}$ = 1 over $2^3$ , keep 2 as the base and change exponent to positive.

12) Use patterns to explain the negative exponent law.

$2^{-1}$ = 1/2 The answer’s denominator is multiplied by the base each time.

$2^{-2}$ = 1/4

$2^{-3}$ = 1/8

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

Yes. ex: $-2^3$ x $-2^2$ = $-2^{3+2}$ = $-2^5$

For different bases, just use BEDMAS to solve. ex: $2^3$ x $3^2$ = 8×9=72.

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

$5^2$ x $5^3$ = $25^5$ .It is wrong because you have to keep the base for product laws. so it would be $5^{2+3}$ = 3125.

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

Order of operation: BEDMAS

ex: 4x $(2)^3$ = 4×8=32.

16) Determine the sum and difference of two powers.

Just use BEDMAS for sums or differences of two powers.

ex: $(6+3)^2$ -21 = $9^2$ -21 = 81-21=60.

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

ex: $(7^5)^2$ = $35^2$ =1225. Should use power law, so multiply exponents, 5×2, which is $7^{10}$ = 282475249.

18) Use powers to solve problems (measurement problems)

$4^2$ – $2^2$

=16-4

=12 $cm^2$

19) Use powers to solve problems (growth problems)

Bacteria doubles every hour. There are 10 bacteria now. How many will there be after each amount of time? 1 hour: 10×2 because it doubles and double means x2. which will be 20 bacteria

2 hour: 10x2x2= 10x $2^2$ = 40 bacteria. 3hours:10x2x2x2=10x $2^3$ =80 bacteria.

11 hours:10x $2^{11}$ = 20480 bacteria. n hours:10x $2^n$ .

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

ex: $2^{-5}$ / $4^{-2}$ = $4^2$ / $2^5$ = 16/32 = 1/2

2 Comments

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wendyhw2016

November 9, 2016 — 6:31 am

Parent comment:
Wendy did a very good job of explaining different exponent laws and the examples are well done. I’m proud of her work.

brookeg2016

November 10, 2016 — 11:01 pm

1) All the learning outcomes are addressed
2) For number 6 there is a rectangle in the question which makes me pretty confused, other than that it’s easy to understand.
3) For number 3 there is no model for $3^2$ and it is kind of hard to visually understand the power but since there is still an explanation I can understand it mentally. In number 9 there is an error with powers, you wrote $5^3$ / $5^3$ = 15/15 when it actually equals 125/125. In number 10 for $-3^5$ the answer is wrong.
4) You explained most questions well. Your usage of examples made it easy to understand.
5) You could use variable bases in your examples. You should double check your work when you finish.

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