Rationalizing expressions (check pins)

[dark lotus]

HELP

[potent zinc]

Where's ur question?

[silent sparrow]

what is the q

[potent zinc]

Op opened a help channel,so ig we r getting none

[dark lotus]

hi

I have more questions

this is first one

@silent sparrow

@potent zinc

please guide me through

I am failing

everything

[potent zinc]

Do u know how to multiply conjugate in both numerator and denominator?

[dark lotus]

no

can you do the first question

and tell me how u did it

and show the answer

cuz i think this hw is about patterns

[marble cliff]

No, people here won't give you an instant answer.

[dark lotus]

I know

show me the working

why r u here again lol

[sonic meadow]

.pin

[marble cliff]

You said that you're going to forum, so I followed you.

[dark lotus]

oh ayt

[dark lotus]

so can you show me how you did the question

using that mutiply way

like u times the base/base

I think

idk

[dense wedge]

Rationalizing expressions (check pins)

[marble cliff]

$\frac{1}{\sqrt{a}+\sqrt{b}} = \frac{\sqrt{a}-\sqrt{b}}{(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b})}$

[gray wharf]

Xwtek

[dark lotus]

is this?

[dark lotus]

okay quick question

how tf do yhou remember this command

$\frac{1}{\sqrt{a}+\sqrt{b}} = \frac{\sqrt{a}-\sqrt{b}}{(\sqrt{a}+\sqrt{b})}$

[gray wharf]

Aden

[dark lotus]

I think is only this

$\frac{1}{\sqrt{a}+\sqrt{b}} = \frac{\sqrt{a}-\sqrt{b}}{(\sqrt{a}+\sqrt{b})\sqrt{a}-\sqrt{b}}$

[gray wharf]

Aden

[dark lotus]

wait no

$\frac{1}{\sqrt{a}+\sqrt{b}} = \frac{\sqrt{a}-\sqrt{b}}{\sqrt{a}+\sqrt{b}$

[gray wharf]

Aden
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[dark lotus]

eh what

I think is smth like this

@marble cliff

[marble cliff]

Yeah, that's the same.

I just fused them into a single fraction.

[dark lotus]

Is this correct? @marble cliff

[marble cliff]

Yes

[dark lotus]

whats the next step

idrk

[marble cliff]

Just multiply the numerators and denominator separately

[dark lotus]

i dont know how tbf

can u do the command thing

[marble cliff]

$\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$

[gray wharf]

Xwtek

[dark lotus]

eh

is this correct?

[marble cliff]

Yes

Now you can expand the denominator

[dark lotus]

why did this become minus?

@marble cliff

[marble cliff]

Because of an important property.
$$(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b})=a-b$$

[gray wharf]

Xwtek

[dark lotus]

eh what

[marble cliff]

You can try expanding the multiplication yourself to verify

[dark lotus]

is this right

[marble cliff]

Yes

[dark lotus]

okay

thats final answer?

[marble cliff]

Yup

[dark lotus]

wait quick question

can u show the working out

for the whole question

with command thingy

[marble cliff]

$$\frac{1}{\sqrt{1}+\sqrt{2}}=\frac{1}{\sqrt{1}+\sqrt{2}}\cdot\frac{\sqrt{1}-\sqrt{2}}{\sqrt{1}-\sqrt{2}}=\frac{\sqrt{1}-\sqrt{2}}{(\sqrt{1}+\sqrt{2})(\sqrt{1}-\sqrt{2})}=\frac{\sqrt{1}-\sqrt{2}}{-1} = \sqrt{2}-1$$

[gray wharf]

Xwtek

[dark lotus]

How does this go to this? @marble cliff does it cancels out with 1 or smth?

[heady root]

Dividing by -1 is the same as multiplying by -1

[dark lotus]

okay

can someone explain this

@heady root @marble cliff

[heady root]

Which equality

[dark lotus]

Why doesthis

go to this

and goes to this

[heady root]

^

[dark lotus]

oh

ok

[heady root]

That’s the first step

And then the second one you just expand

[dark lotus]

can i send yall the working out on paper and see if I get full marks if this is exam?

@marble cliff

@heady root

[heady root]

,rccw

[dark lotus]

huh

is this working out correct?

4 of them

and the answers

@heady root

[heady root]

!noping

[dense wedge]

Please do not ping individual helpers unprompted.

[heady root]

I’m thinking

[dark lotus]

okie

[marble cliff]

Actually it's fine to ping me. But I'm an exception, and you should not ping other helpers.

[dark lotus]

oh

u js werent here

mb

[heady root]

Actually mostly the same lol but in general

[dark lotus]

so can u check my working outs?

[heady root]

The second to last equality is incorrect but then you get the right answer

For 4

[dark lotus]

oh

[heady root]

I don’t see anything wrong with other 3 but I have missed stuff before

[dark lotus]

the 2 - root5?

[heady root]

Yeah maybe you meant -(2-root5)?

[dark lotus]

i need help with these

they are like related

[heady root]

You can just skip to the answer and ignore the 2nd to last equality

[dark lotus]

so erase the 2 - root 5

okay

Can i get help on those 2?

the pattern

i dont see pattern on the answers

1. Root 2 - 1
2. Root 3 - Root 2
3. 2 - Root 3
4. Root 5 - 2

[heady root]

Try writing your answers next to each other but use sqrt 4 instead of 2, and then do the same for sqrt 1

[dark lotus]

Wym

wait

oh

i see it

root 2 - root 2 root 3 - root 2 root 4 - root 3 root 5 - root 4

how tf am I gonna explain that

idk how to explain it

is hard to explain

[heady root]

Use a variable like x to come up with a formula

[marble cliff]

Try to simplify
$$\frac{1}{\sqrt{a}+\sqrt{a+1}}$$

[heady root]

Actually maybe n is more appropriate letter here but you can use whatever

[gray wharf]

Xwtek

[dark lotus]

i js needa write out the pattern

ik the pattern

idk how to explain the pattern

i see the pattern

[heady root]

Try to write the expression above in the form of the answers you got

[dark lotus]

wym by that

[heady root]

Using the same methods you used for the 4 examples

[dark lotus]

ik this is the pattern

[heady root]

This is what the questions came as for a=1,2,3,4 see if you can make a formula for the answers that match up

,rccw

[dark lotus]

like the first sqrt of every answer is like going by 1

and the second sqrt of every answer is going by 1 as well each time

but how do I write it better

[heady root]

Using variables

[dark lotus]

idk how to use variables

can u show me

[heady root]

Like 3x+5=10, variables like x actually this might be confusing

[dark lotus]

huhhhhh

can I say

x is increased by 1 and y is also increased by 1 when i move along the line

[marble cliff]

No

[dark lotus]

how should i say it

i actually stuck

[marble cliff]

Again, look at my hint above.

[marble cliff]

This one

What does it simplify to.

[heady root]

I think Aden might not understand how to use variables in this context

[dark lotus]

i am not sure

[marble cliff]

Use the same steps as you've been doing before.

[dark lotus]

eh okay i will try

[heady root]

This time it’s not numbers you know but the method will still work

[heady root]

Amazing :D

[dark lotus]

is the thingy flipped

these

@heady root but like i still dont know how to explain the pattern lwk

is it like

the answer of the base of the question is fliped

[heady root]

You can say this is equal to what you worked out

[dark lotus]

should

Should I describe the pattern as

idek

how to explain it

[heady root]

If you put the equation down it will describe itself

[dark lotus]

but i need words

the teacher said

[heady root]

Oh okay

[dark lotus]

i understand the whole concept

can u js help me with this describing bit

please

[heady root]

I personally would give a formula full marks but the questions be strict sometimes

Tbh idk how to describe it well

[dark lotus]

eh

[heady root]

The 1/ disappears and the smaller sqrt becomes negative I guess

[heady root]

That phrasing would help with the 2nd question I suppose

[dark lotus]

yep

is it just root 28 - root 27

[heady root]

Yeah

[dark lotus]

Isnt root 28 2 root 7?

[heady root]

Yes, might be less clear you got the right answer though

You can do the same with 27

[dark lotus]

oh ok

[heady root]

Perhaps put both down simplified and unsimplifed

[dark lotus]

i can js put root 28 - root 27?

[heady root]

Yeah that and also the simplified one

$\sqrt{27}=3\sqrt{3}$

[gray wharf]

BBMaths

[dark lotus]

okay

is it just this?

[heady root]

Oh yeah lol

Use n instead of a since they ask for it

[dark lotus]

what the fuck is this omdsss

@heady root

[heady root]

Use the formula you made

It will simplify nicely

[dark lotus]

wait what

which formula

[heady root]

The one you just wrote down

[dark lotus]

can u show me question one and explain how u did it

[heady root]

Just put a=1 and a=2 into these formulas and rewrite each term like that

[dark lotus]

oh ok

whats this @heady root

[marble cliff]

context?

[heady root]

That’s the answer

Not sure what the red \ is doing

[dark lotus]

whats this @heady root

[marble cliff]

As in infinite series?

[dark lotus]

i am not sure what it mean

s

[marble cliff]

That's a weird way to write a finite summation.

[dark lotus]

so how do I answer this question

[marble cliff]

$$\frac{1}{\sqrt{1}+\sqrt{2}} +\frac{1}{\sqrt{2}+\sqrt{3}} + ... + \frac{1}{\sqrt{k}+\sqrt{k+1}} = (\text{fill this yourself})$$

[gray wharf]

Xwtek

[marble cliff]

Tip: use telescoping summation.

[dark lotus]

no idea what this is

[gray wharf]

Aden

[dark lotus]

nah but like these are related I think @marble cliff

[marble cliff]

By the way, have you found the answers for the first three question?

[dark lotus]

i will send u

@marble cliff here u go

[marble cliff]

I think the $\frac{1}{\sqrt{3}+\sqrt{4}}$ part is incorrect.

[gray wharf]

Xwtek

[dark lotus]

question 2?

[marble cliff]

Question 2 and 3

[dark lotus]

kk

now?

also how to do that question after

@marble cliff

[marble cliff]

Then you learn about telescoping sums:
$$(f(2)-f(1))+ (f(3)-f(2))+(f(4)-f(3))+...+(f(n+1)-f(n))=f(n+1)-f(1)$$

[gray wharf]

Xwtek

[dark lotus]

i continuee the pattern

,rccw

[marble cliff]

I'm sure you have to solve the pattern for any k, not just until 13

[dark lotus]

how

i give up on this hw

[marble cliff]

Again, simplify this.

[dark lotus]

i cant bro

i give up

[arctic widget]

Let's try another approach

$\frac{1}{\sqrt1 + \sqrt2}$

[gray wharf]

This is sad 😢

[arctic widget]

Can you rationalize this for me?

[marble cliff]

It has been solved above

[arctic widget]

uhhhh

well, then lemme check what's the problem