Question game

Hii everyone. This note is a type of quiz game where you can solve questions and post them too. The advantage of this note is you can practice both solving and posting good questions. Your solutions and questions will be appreciated too by community members. So, enjoy the beauty of solving the questions. Go ahead :

Rules And Regulations :

  1. In this game the questions will be posted and by the community members itself.

  2. The one who solves the latest question first will be able to post a new question. You should provide a clear solution to the question you have solved. Only the one who first solves the latest question correctly will have a chance to post the new question.

  3. If any member has a different solution to a question which has already been solved he can also post the solution but he cannot post a new question.

  4. There is also a leader board (below) in which the top 10 highest solvers will be shown. So try to solve as many questions as possible.

  5. The one who posted the question should not be the first one to solve that question. However he can post his solution if he wants only after anybody solves his question.


Summary

Total Questions Solved : 3

Latest Question Posted : Problem 3

Leader Board (Top 10 highest Solvers) :

NAME - PROBLEMS SOLVED

  1. Vaibhav Priyadarshi and Vilakshan Gupta and Michael Mendrin - 1

  2. Ram Mohith and R Mathe - 1

  3. Matin Naseri - 1


If any improvements are required in this game kindly inform them to me.

Thank You and All the Best

Also see my proof based notes in my set Proof Based Notes

Note by Ram Mohith
1 month ago

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1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 \( 2 \times 3 \)
2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)

Comments

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Top Newest

I think this game lacks number of solvers

Ram Mohith - 2 days, 15 hours ago

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Yes,Maybe. but How to we should solve the issue?

Matin Naseri - 2 days, 13 hours ago

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Yes, probably.

Michael Mendrin - 2 days, 15 hours ago

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If many people start to participate then this game would be really creative.

Ram Mohith - 2 days, 3 hours ago

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76 members have grown this note but only 6 people are participating.

Ram Mohith - 2 days, 3 hours ago

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\(\text{Problem 4:}\)


Two unit disks are randomly thrown on the plane. If they overlap, what's the expected distance between their centers?

Michael Mendrin - 3 weeks, 5 days ago

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I know two methods of doing it, but I know none!

One method is by using an excel sheet and other is by integration (both of which I don't know)

Just another guess....0.67

Vilakshan Gupta - 3 weeks, 1 day ago

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Yes, the correct answer is 2/3

Michael Mendrin - 3 weeks, 1 day ago

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@Michael Mendrin Sir,you may ask another question since this was solved by none...

Vilakshan Gupta - 3 weeks ago

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1

Vaibhav Priyadarshi - 3 weeks, 5 days ago

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The answer is \(\boxed{0}\).

Tom Clancy - 3 weeks, 4 days ago

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@Tom Clancy The minimum and maximum possible distances are 0 and 2, so the expected value is somewhere inbetween.

Michael Mendrin - 3 weeks, 4 days ago

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@Michael Mendrin Okay, thus I think the final answer is \(\boxed{0.5}\).

What about \(\boxed{0.5}\)?

Tom Clancy - 3 weeks, 4 days ago

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@Tom Clancy In mathematics, you don't get to guess. But, I can tell you it's some kind of a fraction.

It's not 0.5 and it's not 1/2

Pick a fraction between 0 and 2.

Michael Mendrin - 3 weeks, 4 days ago

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I solved it like this: If they overlap, maximum distance between their centres would be 2(if they just touches each other). Minimum distance between their centres would be 0, when they are coincident. So expected distance can be average of maximum and minimum which is 1.

Vaibhav Priyadarshi - 3 weeks, 5 days ago

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@Vaibhav Priyadarshi Okay, well, no, 1 is not the correct answer. In fact, if you do a computer simulation, randomly throwing unit discs on the plane, you won't get 1.

Michael Mendrin - 3 weeks, 5 days ago

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Ah, that's what's wrong with this problem, if I say your answer is wrong, then somebody else will probably figure out the correct answer.

Michael Mendrin - 3 weeks, 5 days ago

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Two unit disks are randomly thrown on the plane. If they overlap, what's the expected distance between their centers?

Michael Mendrin - 3 weeks, 5 days ago

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\(\text{Problem 3:}\)


In the \(\triangle ABC\) ,\(O\) is any arbitrary point in the interior of the triangle. Lines \(OA\) , \(OB\) , \(OC\) are drawn such that the angles \(OAB\),\(OBC\) and \(OCA\) are each equal to \(\omega\) , then prove that \[\cot A+\cot B+\cot C=\cot \omega\] and \[\csc^2A+\csc^2B+\csc^2C=\csc^2\omega\]

Vilakshan Gupta - 1 month ago

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Let \(a,b,c\) be the angles of the triangle and \(x\) be the angle common to all 3 vertices. Then using Law of Sines, we have

\(\dfrac{Sin(x)}{OB}=\dfrac{Sin(b-x)}{OA}\)
\(\dfrac{Sin(x)}{OC}=\dfrac{Sin(c-x)}{OB}\)
\(\dfrac{Sin(x)}{OA}=\dfrac{Sin(a-x)}{OC}\)

Hence

\(4{ \left( Sin(x) \right) }^{ 3 }=4Sin(a-x)Sin(b-x)Sin(c-x)\)

And from there on, using product-to-sum and sum-to-product trig identities, we have

\(3Sin(x)-Sin(3x)=Sin(2a+x)+Sin(2b+x)-Sin(2(a+b)-x)-Sin(3x)\)

then after rearrangement and adding the term \(-Sin(2a-x)\) on both sides

\(-Sin(2a-x)+Sin(x)+Sin(2(a+b)-x)-Sin(2b+x)=-Sin(2a-x)-2Sin(x)+Sin(2a+x)\)

then

\(-4Cos(x)Sin(a)Sin(b)Sin(a+b)+4Cos(a)Sin(b)Sin(a+b)Sin(x)=\) \(-4Cos(b)Sin(a)Sin(a+b)Sin(x)+4Cos(a+b)Sin(a)Sin(b)Sin(x)\)

and finally

\(Sin(a)Sin(b)Sin(c)Sin(x)(Cot(a)+Cot(b)+Cot(c)-Cot(x))=0\)

Michael Mendrin - 1 month ago

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Can you explain your first step, from where did the first equation come?

Vilakshan Gupta - 1 month ago

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@Vilakshan Gupta I'll put that first step in, see my solution

Michael Mendrin - 1 month ago

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@Michael Mendrin Vilakshan Gupta is Micheal Mendrin solution correct or not. Please tell I have to update the note.

Ram Mohith - 1 month ago

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@Ram Mohith Ram, the full solution is going to be much longer than what I've posted, many more steps inbetween. What I've posted are the key steps, especiallly the middle part. I've not been able to figure out a shorter way to prove it. Maybe Gupta can just post a new question and start over.

Michael Mendrin - 1 month ago

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@Michael Mendrin The solution seems fine, mine is also a similar solution. Then according to the rules of this game, you can post a new question Sir.

Vilakshan Gupta - 4 weeks ago

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@Vilakshan Gupta Ok.Now the question is solved by Michael Mendrin sir and he can now post the latest questions.

Ram Mohith - 4 weeks ago

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Good solution sir. If it is correct you are the one who can post the next question. But let's see what Vilaskhan Gupta says.

Ram Mohith - 1 month ago

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Ok. It is tough but no problem. But from next time onwards try to post problems of Easy and Medium difficulty.

Ram Mohith - 1 month ago

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It's creative. I want to join into your discussion.

Matin Naseri - 1 month ago

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You can. In fact I am encouraging everyone to join this. Try to solve any question the first and you can post a new question.

Ram Mohith - 1 month ago

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\(\log_2 10 - \log_8 125\)

\(\log_8 125 ={\log_{2^3} 125}\)

\(\dfrac{1}{3}{\log_{2}125}\), \(\sqrt[3]{125}=5\)

Thus \(\log_2 5 ={\log_8 125}\)

\(\log_2 10 - \)\(\log_2 5 = \)\(\log_2{\frac{10}{5}}=\)\(\log_2 2 = 1\)

Matin Naseri - 1 month ago

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@Matin Naseri Fine but you are the third solver to that question

Ram Mohith - 1 month ago

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@Ram Mohith What I should do?

Also I have fixed my code,My last line had Errors.

Matin Naseri - 1 month ago

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@Matin Naseri No problem. Leave your solution. It will come into your account. If you solved another question you will have 2 solutions and will be top in the leader board. Try to solve the latest question : Problem - 3

Ram Mohith - 1 month ago

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\(\textbf{Problem 2 :}\)

Find the value of \((5+2\sqrt{13})^{1/3}+(5-2\sqrt{13})^{1/3}\)

Vaibhav Priyadarshi - 1 month ago

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Consider an arbitrary polynomial \(X^{3}+aX+b\in\mathbb{R}[X]\). Then, provided \(\Delta:=(a/3)^{3} + (b/2)^{2} > 0\), letting

\[C:=-b/2+\sqrt{\Delta},\quad D:=-b/2-\sqrt{\Delta}\]

one has (taking real cubed roots) that \(C^{1/3}+D^{1/3}\) is the unique real root of \(X^{3}+aX+b\).

Now choosing \(b:=-10\) and \(a:=9\), one has \(5+2\sqrt{13}=5+\sqrt{52}=C\) and \(5-2\sqrt{13}=5-\sqrt{52}=D\) (one may obtain the values of \(a,b\) by simply solving \(-b/2=5\) and \(\Delta=52\). Thus the expression in the problem, \(\breve{x}:=(5+2\sqrt{13})^{1/3}+(5-2\sqrt{13})^{1/3}\), is equal to the unique real root of

\[X^{3}+9X-10\]

Now, note that \(1\) is a root of this polynomial. Thus it is the unique real root. Hence \(\boxed{\breve{x}=1}\).

R Mathe - 1 month ago

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Good. Now you too solved this question but you cannot post the new question because Vilakshan Gupta already solved it. But this solution comes into your account.

Ram Mohith - 1 month ago

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@Ram Mohith @Ram Mohith But this solution comes into your account. I have no idea what this sentence means. Regardless. I solved it for the sake of solving it. I don’t pay attention to who does what first nor do I look at other people’s solutions until I’ve done things myself.

@Michael Mendrin I worked completely in \(\mathbb{R}\). For odd \(n\in\mathbb{N}\) the unique real \(n\)th root of \(-a\) does coincide with \(-(a^{1/n})\). I do appreciate that if we work in \(\mathbb{C}\) it is all a matter of convention: there is no compelling reason to choose one argument over another (ie the choice of \(k\in\mathbb{Z}\) in \(\frac{\theta+2\pi k}{n}\)).

R Mathe - 1 month ago

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@R Mathe Well, right, what do we do if we have multiple values for this sort of thing? Just like just because we know that \((-2)(-2)=4\) doesn't mean it's safe to assume that \(\sqrt{4}=-2\). Likewise fractional powers of negative numbers. Whenever I see something like that arise, it's never safe to "convert a complex into a real".

What's the value of \((-1)^{\frac{1}{3}}\)?

I meant to explore this more in detail this morning, but this Question Game moves on.

Michael Mendrin - 1 month ago

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@Michael Mendrin Your example with even values is irrelevant to what I was saying. I referred only to odd \(n\in\mathbb{N}\). For odd powers there is only one real root.

For \(n\in\mathbb{N}\) odd, the map \(f:x\in\mathbb{R}\mapsto x^{n}\in\mathbb{R}\) is continuous and unbound above and below, thus the image (as a continuous image of a connected set) is all of \(\mathbb{R}\), moreover the map is injective. Hence there is a unique Inverse in \(\mathbb{R}\). Thus for all \(a\in\mathbb{R}\) there is exactly one value \(b\in\mathbb{R}\) such that \(b^{n}=a\). This is the unique real \(n\)th root. Eg, the unique real \(3\)rd root of \(-1\) is \(-1\).

it's never safe to "convert a complex into a real"

I disagree with this. What is sensible is it is never safe to not fix a framework. And fix a framework I did. There is no compulsion to work in one particular field. Since eg \(\mathbb{Q}\subseteq\mathbb{R}\subseteq\mathbb{C}\subseteq\mathrm{GL}(\mathbb{C},3)\subseteq\ldots\) are field extensions, one could by your logic also argue, that it’s never safe to work in \(\mathbb{C}\) and that one should always work in some \(\mathrm{GL}(\mathbb{C},3)\).

R Mathe - 1 month ago

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@R Mathe Let me try to explain what I think the problem is. Suppose instead of something as complicated as the original expression, the problem asks, "what's the value of \((-1)^{\frac{1}{3}}\)"? There is no one unique answer, unless the problem asks for the unique real root. If one argues that "1" is a valid value of the original complicated expression, I can equally argue that my complex value is equally valid.

What I am saying, in ordinary mathematical computations, one must be careful about assuming only the unique real roots of such fractional powers of negative quantities. Otherwise, I could "show" that \(-1 = \frac{1}{2} +\frac{\sqrt{3}}{2}i\), for example (and therefore -1= 3!)

Michael Mendrin - 1 month ago

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@Michael Mendrin The value of \((-1)^{\frac{1}{3}}\) should be \((-1)^{\frac{1}{3}}\), which denotes a multiple-valued complex quantity. It's kind of analogous to some quantities in quantum physics, huh? "It can have a range of values, depending on how you look at it".

Michael Mendrin - 1 month ago

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@Michael Mendrin It is not a multivalued object (except maybe in set theory). In algebra, it’s just ill-defined. And in complex analysis one acknowledges the existence of many inverses, and the consequences of this being any choice of the \(n\)th root function \[f:z\in\mathbb{C}\mapsto \sqrt[n]{|z|}\exp(\imath\frac{\arg(z)+2\pi k}{n})\] is holomorphic except along a semi-infinite line.

R Mathe - 1 month ago

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@R Mathe "In algebra, it' s just ill-defined" (!). Well, isn't that what I've been trying to argue?

Why is my value \(\frac{1}{4}(1+3\sqrt{13}+i(\sqrt{39}-\sqrt{3}))\) any less valid than "1"?

Michael Mendrin - 1 month ago

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@Michael Mendrin Being ill-defined here means there be no single-valued (inverse) function \(f:\mathbb{C}\longrightarrow\mathbb{C}\). Being multi-valued means a function of the form \(f:\mathbb{C}\longrightarrow\wp(\mathbb{C})\). Both of these are technically true. But in algebra, it is not sensible to work with the latter.

Your result is fine. The whole point of this discussion, was you disputing mine. I defended my result by stating that I’m working in the framework \(\mathbb{R}\). You’re working in the framework \(\mathbb{C}\). Which framework one chooses is another issue. As soon as these are fixed, the answers are what they are—within the framework \(\mathbb{R}\) the only valid result is \(1\). In your framework, which is perfectly legitimate your result is one of several perfectly valid answers.

R Mathe - 1 month ago

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@R Mathe Oh, I don't know how you got the idea that I was disputing your solution. My apologies. Yes, earlier, before you jumped in with your solution, I did blurt out, "let's make sure if 1 is right!" Well, it was late at night, and so I didn't get the chance to review the matter until this morning, and saw what the problem was.

This is not the first time I've run to this sort of thing in Brilliant.org, where powers of complex numbers are involved. Several problems have already been posted to explore this topic.

An annoyance that I have is that for the sake of rigor, a function of the form \(y=f(x)\) is, by convention, defined to be single valued. This is the reason why \(y=\sqrt{x}\) by convention only yields positive \(y\), i.e. only the top half of the parabola. But lots of implicit functions of the form \(f(x,y)=0\) are multi-valued in both the \(x\) and \(y)\) directions. This is the reason why whenever complex quantities are involved, especially powers of them, it raises red flags for me and I move with caution, and ask myself, "okay, what's the convention with this, and in what context is this? " This is just from habit now.

Michael Mendrin - 1 month ago

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@Michael Mendrin Not a problem. Generally any randomly chosen implicit functions will have many locally well-defined branches. There’s no mathematical preference for any. By contrast there is sensible preference however, when one demands restricting a solution to say a particular field. Nonetheless, as I said, one always needs to fix a framework and work in it. If things happen to have more solutions in a larger framework, then so be it. For example asking, whether \(X^{2}+X+1\) be irreducible without specifying an algebraic structure, is simply begging for problems, as the answers is yes or no, depending on the context (if the field is \(\mathbb{R}\), \(\mathbb{Q}\), \(\mathbb{C}\), \(\mathbb{F}_{2}\), … or if one works in a ring, etc.).

R Mathe - 1 month ago

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@Michael Mendrin Sir, please try to solve the latest problem 3

Ram Mohith - 1 month ago

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@Ram Mohith If we rewrote the expression to \((5+2\sqrt{13})^{\frac{1}{3}}-(-5+2\sqrt{13})^{\frac{1}{3}}\), then, yes, the value becomes \(1\). But what is done is to assume that \((-a)^{\frac{1}{n}}=-(a)^{\frac{1}{n}}\), which is not true for all \(n>1\). That's not a safe step to take when doing computations, because what that does is to "convert a complex quantity into a real".

This is a controversal subject, involving fractional powers of complex numbers, which are multi-valued, and this sort of thing crops up on Brilliant from time to time.

Michael Mendrin - 1 month ago

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Vaibhav shall I update the note as Vilakshan Gupta solved your question. What do you think?

Ram Mohith - 1 month ago

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Comment deleted 1 month ago

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@Vaibhav Priyadarshi First, how about making sure "1" IS the correct answer !

Michael Mendrin - 1 month ago

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@Michael Mendrin Well, make sure that indeed it is

Vilakshan Gupta - 1 month ago

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@Michael Mendrin Ok sir. After you have confirmed I will update the note

Ram Mohith - 1 month ago

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@Ram Mohith The answer I posted is the correct answer, and I will check again in the morning.

Michael Mendrin - 1 month ago

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The answer is 1.

Vilakshan Gupta - 1 month ago

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How is it "1"? This expression is a root of a 4th degree polynomial with no real roots.

Michael Mendrin - 1 month ago

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@Michael Mendrin Let \((5+2\sqrt{13})^{1/3}\) + \((5-2\sqrt{13})^{1/3}\) = \(x\)

Cubing both sides, we get

\(10+3(5^2-(2√13)^2)^{1/3}x\) = \(x^3\)

\(10-9x=x^3\)

It have 1 as a solution.

Vaibhav Priyadarshi - 1 month ago

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@Vaibhav Priyadarshi \( (5^2-(2\sqrt{13})^2))^{\frac{1}{3}} \) is complex. You slipped up on sign. Or at least about fractional powers of \(-1\).

Let me try to see if I can save this....

Michael Mendrin - 1 month ago

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@Michael Mendrin Is \((-27)^{1/3}\) not real?

Vaibhav Priyadarshi - 1 month ago

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@Vaibhav Priyadarshi Unfortunately, no. This is where it gets complicated.

Michael Mendrin - 1 month ago

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@Michael Mendrin I agree that the principal root of \((-27)^{\frac13}\) is an imaginary number , but we are concerned about the real valued root , and that is , \(-3\).

Check this

Vilakshan Gupta - 1 month ago

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@Michael Mendrin Please explain me why.

Vaibhav Priyadarshi - 1 month ago

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@Vaibhav Priyadarshi Also I didn't understand how you got 9x

Ram Mohith - 1 month ago

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@Ram Mohith By using \(a^3+b^3+3ab(a+b)\)

Here a+b=x.

Vaibhav Priyadarshi - 1 month ago

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@Vaibhav Priyadarshi What about ab. When they both are multiplied they give a complex number

Ram Mohith - 1 month ago

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@Vaibhav Priyadarshi Why have you posted the solution when you have posted the question

Ram Mohith - 1 month ago

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@Ram Mohith Sir have asked me how it is 1, so I have given the reason.

Vaibhav Priyadarshi - 1 month ago

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@Vaibhav Priyadarshi But the problem now is who solved this question and who will post the next one.

Ram Mohith - 1 month ago

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@Ram Mohith Vilakshan Gupta

Vaibhav Priyadarshi - 1 month ago

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Yes it is correct!

Vaibhav Priyadarshi - 1 month ago

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\(\frac{1}{4}(1+3\sqrt{13}+i(\sqrt{39}-\sqrt{3}))\)?

Michael Mendrin - 1 month ago

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Can you give explanation for it? Why isn't this a real number? Principal Cube root of real number should be real.

Vaibhav Priyadarshi - 1 month ago

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@Vaibhav Priyadarshi First tell is it correct or not

Ram Mohith - 1 month ago

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@Ram Mohith I think not.

Vaibhav Priyadarshi - 1 month ago

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@Ram Mohith A lot of algebra went into this, but i can't get rid of the complex part.

Michael Mendrin - 1 month ago

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@Michael Mendrin Let me try another way to do this

Michael Mendrin - 1 month ago

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Sir, if this is correct you can post a new question but whether it is correct or not Vibhav should tell it.

Ram Mohith - 1 month ago

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@Ram Mohith I don't think the answer is "1"

Michael Mendrin - 1 month ago

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@Michael Mendrin I will give you a hint , note that \(a^3+b^3=10\). Rest is upto you

Vilakshan Gupta - 1 month ago

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@Vilakshan Gupta \((a^3+b^3)=(a+b)(a^2-ab+b^2)=10\), but I don't see where the "1" comes up. Both factors are complex conjugates.

Michael Mendrin - 1 month ago

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@Michael Mendrin Actually what I meant was Vaibhav's solution only, because if you cube the expression, the radicals get cancelled off.

If you let the expression to be equal to \(x\) and then cube it, then you will get the equation \(10-9x=x^3\) and from here, by observation , we can conclude that the answer has to be \(1\).

The other two roots have to be complex conjugates, but the answer in real numbers is \(1\).

Vilakshan Gupta - 1 month ago

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@Vilakshan Gupta Ok. Now I understood. So you have solved it Vilakshan Gupta

Ram Mohith - 1 month ago

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@Vilakshan Gupta Vilakshan Gupta why can't you post your solution if you have solved

Ram Mohith - 1 month ago

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@Michael Mendrin Just manipulate the expression using algebra and you will see that...wow... it turns out to be 1!

Some things look complicated but just do the math and you will get the answer.

Vilakshan Gupta - 1 month ago

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Another solution for Problem 1 :

\(\log_8 125 = \log_{2^3} 125 = \dfrac{1}{3}\log_2 125 = \log_2 \sqrt[3]{125} = \log_2 5\)

\(\log_2 10 - \log_8 125 = \log_2 10 - \log_2 5\)

\(\implies \log_2 \dfrac{10}{5} = \log_2 2 = 1\)

Ram Mohith - 1 month ago

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Problem 1 (solved by Vaibhav Priyadarshi)

Find the value of \(\log_2 10 - \log_8 125\)

Ram Mohith - 1 month ago

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\(log_{2}10-log_{8}125\)

=\(log_{2}10-log_{2^3}125\)

=\(log_{2}10-(log_{2}125)/3\)

=\((3log_{2}10-log_{2}125)/3\)

=\((log_{2}10^3-log_{2}125)/3\)

=\((log_{2}(1000/125))/3\)

=\((log_{2}8)/3\)

=3/3 =1.

Vaibhav Priyadarshi - 1 month ago

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Please use \log in your solution.

Ram Mohith - 1 month ago

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Now can I post a new question?

Vaibhav Priyadarshi - 1 month ago

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@Vaibhav Priyadarshi Yes you can post a new question

Ram Mohith - 1 month ago

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