I'm in 9th grade and i'm confused about this math problem. We're solving a division problem with scientific notation.

[AlexandriaSVH]

Part 1
Sylvia, Dylan, and Ethan are solving a division problem with scientific notation.

Sylvia	Dylan	Ethan
3.61?10−114.1?1073.61x10−114.1x107 = .88?10−18.88x10−18	3.61?10−114.1?1073.61x10−114.1x107 = .88?10−18.88x10−18	3.61?10−114.1?1073.61x10−114.1x107 = .88?10−4.88x10−4
= 8.8?10−178.8x10−17	= 8.8?10−198.8x10−19	= 8.8?10−58.8x10−5

1. Explain Sylvia’s processes. Is Sylvia correct? If not where did she go wrong?
2. Explain Dylan’s processes. Is Dylan correct? If not where did he go wrong?
3. Explain Ethan’s processes. Is Ethan correct? If not where did he go wrong?
   

Part 2
Skyler, Robert, and Kaitlyn all solve the same exponential problem but have different approaches.

Skyler	Robert	Kaitlyn
729?=9729x=9	729?=9729x=9	729?=9729x=9
(36)?=9(36)x=9	(93)?=91(93)x=91	(36)?=32(36)x=32
6x = 9	3x = 1	6x = 2
x = 9696	x = 1313	x = 2626
x = 3232		x = 1313

1. Explain Sklyer's processes. Is Skyler correct? If not, where did she go wrong
2. Explain Robert's processes. Is Robert correct? If not, where did he go wrong?
3. Explain Kaitlyn's processes. Is Kaitlyn correct? If not, where did she go wrong?

 

[Dr.Peterson]

Part 1
Sylvia, Dylan, and Ethan are solving a division problem with scientific notation.

Sylvia	Dylan	Ethan
3.61?10−114.1?1073.61x10−114.1x107 = .88?10−18.88x10−18	3.61?10−114.1?1073.61x10−114.1x107 = .88?10−18.88x10−18	3.61?10−114.1?1073.61x10−114.1x107 = .88?10−4.88x10−4
= 8.8?10−178.8x10−17	= 8.8?10−198.8x10−19	= 8.8?10−58.8x10−5

1. Explain Sylvia’s processes. Is Sylvia correct? If not where did she go wrong?
2. Explain Dylan’s processes. Is Dylan correct? If not where did he go wrong?
3. Explain Ethan’s processes. Is Ethan correct? If not where did he go wrong?

Part 2
Skyler, Robert, and Kaitlyn all solve the same exponential problem but have different approaches.

Skyler	Robert	Kaitlyn
729?=9729x=9	729?=9729x=9	729?=9729x=9
(36)?=9(36)x=9	(93)?=91(93)x=91	(36)?=32(36)x=32
6x = 9	3x = 1	6x = 2
x = 9696	x = 1313	x = 2626
x = 3232		x = 1313

1. Explain Sklyer's processes. Is Skyler correct? If not, where did she go wrong
2. Explain Robert's processes. Is Robert correct? If not, where did he go wrong?
3. Explain Kaitlyn's processes. Is Kaitlyn correct? If not, where did she go wrong?

It looks like you pasted this in and didn't proofread. Don't you see how badly it is messed up?

Here is what I get if I just remove duplicates, and change some numbers into exponents, and insert divisions:

Part 1
Sylvia, Dylan, and Ethan are solving a division problem with scientific notation.

Sylvia	Dylan	Ethan
3.61x10−11/4.1x107 = .88x10−18	3.61x10−11/4.1x107 = .88x10−18	3.61x10−11/4.1x107 = .88x10−4
= 8.8x10−17	= 8.8x10−19	= 8.8x10−5

1. Explain Sylvia’s processes. Is Sylvia correct? If not where did she go wrong?
2. Explain Dylan’s processes. Is Dylan correct? If not where did he go wrong?
3. Explain Ethan’s processes. Is Ethan correct? If not where did he go wrong?

Part 2
Skyler, Robert, and Kaitlyn all solve the same exponential problem but have different approaches.

Skyler	Robert	Kaitlyn
729x=9	729x=9	729x=9
(36)x=9	(93)x=91	(36)x=32
6x = 9	3x = 1	6x = 2
x = 9/6	x = 1/3	x = 2/6
x = 3/2		x = 1/3

1. Explain Sklyer's processes. Is Skyler correct? If not, where did she go wrong
2. Explain Robert's processes. Is Robert correct? If not, where did he go wrong?
3. Explain Kaitlyn's processes. Is Kaitlyn correct? If not, where did she go wrong?

Check whether I got everything right (I think I missed several things!), and then try showing us some of your own thinking. What do you think the correct answer to each problem is, and what do you think the others might have been thinking?

 

[AlexandriaSVH]

This is the correct one! i'm so sorry, copying and pasting make things mess up SO BAD clearly?‍, and i know that
Skyler is correct at the beginning, but Skyler makes a mistake
3^6 does equal = 729
so this is correct
(3^6)^x = 9
You can only equate exponents if both sides have
the same base. Here, Skyler is mixing exponents
and bases .
thenfrom here ,
she could say.... 3^(6x) = 9
then, since 9 = 3^2
3^(6x) = 3^2
SInce the bases are now equal,
the exponent must be equal

6x = 2
x = 2/6 = 1/3 BUT i turned that in and i needed extra details and became confused, I couldn't explain Sylvia,Ethan's Nor Roberts that's probably the part i don't understand enough to explain why they're correct or not .

 

[Dr.Peterson]

Skyler is correct at the beginning, but Skyler makes a mistake
3^6 does equal = 729
so this is correct
(3^6)^x = 9
You can only equate exponents if both sides have
the same base. Here, Skyler is mixing exponents
and bases .
thenfrom here ,
she could say.... 3^(6x) = 9
then, since 9 = 3^2
3^(6x) = 3^2
SInce the bases are now equal,
the exponent must be equal

6x = 2
x = 2/6 = 1/3 BUT i turned that in and i needed extra details and became confused, I couldn't explain Sylvia,Ethan's Nor Roberts that's probably the part i don't understand enough to explain why they're correct or not .

Yes, this is the correct answer; so Kaitlyn is correct, right?

So what is Robert doing? He got the right answer, but used a different base. Is his work valid? (In all three cases, I'd have written an extra line before the third row, in this case \(9^{3x}=9^1\) -- just as you did in your work.)

Then you say you don't understand Sylvia and Ethan; does that imply you think Dylan is right? If so, good.

Do you see where each of the others first differs from that correct answer? How would each get their wrong line from the one before?

The value in this exercise is to help you catch your own errors, by looking closely at details like signs; one good way to find them (and to prevent errors of your own) is to write extra steps. For example, in Dylan's work you might start out with \(0.88\times 10^{-11-7}\). That makes it clearer what should come next.