ExponentsHUBBARD2017 – Katya's Blog

Everything I know about Exponents

1.Repeated multiplication with exponents:

Example: (2)(2)(2)(2)(2)= $2^5$ = 32

How I did it: The base is the number that is getting written down (in this case 2). The exponent is the number of times the number is written down/multiplied (in this case 5)

2.How powers represent repeated multiplication:

Example: (3)(3)(3)= $3^3$ = 27

How and why I did it: The base is the number that is getting written down (in this case 3). The exponent is the number of times the number is written down/multiplied (in this case 3) Its a way easier way of showing it instead of writing the whole thing out (3x3x3)

3.Difference between= $2^3$ and $3^2$

Left image: Since its 2 cubed, we know that its going to be a 3D shape so were looking for the volume. (2x2x2)

Right Image: Since its 3 squared, we know that its going to be a 2D shape so were looking for the area. (3×3)

4. $2^3$ is 2x2x2 =8 and $3^2$ is 3×3 =9

5. $4^5$ is 4x4x4x4x4= 1024 $5^4$ is 5x5x5x5= 625

6. $(-2)^4$ = (-2)(-2)(-2)(-2) = 16

( $-2^4$ ) = -1x2x2x2x2 = -16

$-2^4$ = -1x2x2x2x2 = -16

7. Multiplying exponents with same base:

$2^3$ x $2^2$ = $2^5$ How: You keep the base but add the exponents

Dividing exponents with same base:

$3^8$ : $3^4$ = $3^4$ How: You keep the same base but subtract the exponents

8. Exponent laws for multiplying and dividing powers with the same base

When the base is the same, you only have to change the exponents. (add when multiplying and subtract when dividing)

9. Law for power rules when exponent is 0

When the exponent is 0, the answer is going to be 1

10. $2^4$ = 16, $2^3$ = 8, $2^2$ = 4, $2^1$ = 2, $2^0$ =1

It divides by 2 each time

11. Law for powers with negative numbers:

You simply flip the negative fraction and change the negative exponent to positive. In this case: 5 to the exponent of -3, is 5/1 to the exponent of -3. But if you flip it, the answer will be 1/5 to the exponent of positive 3

12. Patterns to explain the negative exponent law:

The pattern is, is that it divides by 2 each time. So when you reach 1, 1 divided by 2 is 1/2, 1/2 divided by 2 is 1/4, and 1/4 divided by 2 is 1/8 and so on

13. Exponent laws to powers with both integral and variable bases

Integral base- Since the top and bottom are negative, they automatically become positive so it will just be 3. Then 3-1 is 2 so it become 3 to the power of 2. The answer is 9

Variably base- -3+-2 is -5 so the answer is x to the exponent of -5

14. Error in a simplification of an expression involving powers

$2^7$ x $2^2$ = $2^8$

This is wrong because 7+2 is not 8. It is 9. Correct answer: $2^9$ . So the incorrect part was the addition of the exponents.

15. Order of operations on expressions with powers

Using BEDMAS:

Brackets, exponents, division, multiplication, addition, and subtraction

In this case: Since there are no brackets we first do the exponents. $2^3$ is 8. The equation is now 8+4 divided by 2+-9. The division part comes second. (4 divided by 2 is 2) Then do the rest left to right. 8+2+-9=10+-9= 1.

16. Sum and difference of two powers

Sum (do the exponents first then add)- $3^2$ + $2^5$ = 9+32= 41

Difference (do the exponents first then subtract)- $9^2$ – $4^2$ = 81-16= 65

17. Error in applying order of operations in an incorrect solution:

The one on the left was wrong because BEDMAS wasn’t used correctly. They just did everything left to right. Correct equation and how to do it on the right. (following correct BEDMAS rules!)

18. Use powers to solve measurement problems:

You add the two squares together by:

5×5+2×2 which is 25+4=29cm2

(make sure you put units! In this case it was in cm and since its 2D, make sure its squared as well!)

19. Use powers to solve growth problems:

Group of bacteria triples every hour. There are 200 bacteria now, how many will there be after 3 hours?

after 2- $200^3$ = 800

after 3- $800^3$ = 51200

Since there are 200 at first, 200×3=800. (x3 sine we want after 3 hours) 800 to the exponent of 3 is 51200. (to the exponent of 3 since it triples every hour)

20. Order of operations on expressions with powers involving negative exponents and variable bases:

Explanation: You simply start at the top. a to the exponents of -5 times -3 so a to the exponent of 15. Then b to times -3 so b to the exponent of -3. Then same with the bottom. a to the power of 4 times -2 is equal to b to the power of -8 and then b times the power of -2 is b to the power of -2.

15- (-8) is 23 so a to the power of 23 on the top. -3 -(-2) is b to the power of -1.