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x⁵ = 7

Taking log both sides: (ln = log_e)

ln x⁵  = ln 7   (ln 7 = 1.94591)

5 ln x = 1.94591   (ln x^y = y ln x)

ln x = 1.94591

               5

ln x = 0.389182

Taking antilog:

x = e^0.389182

x = 1.475773162

x^5=7

\(x^5=7 \\ x = \sqrt[5]{7}\\ x = 1.47577316159\)

eliminate "vi" from the following equations 1)vf - vi + at S=vit+1/2at^2

\(\begin{array}{lrcl} (1) & v_f &=& v_i + a\cdot t \\ & v_i &=& v_f - a\cdot t \\\\ (2) & s &=& v_i \cdot t +\frac12 a\cdot t^2 \\ & s &=& (v_f - a\cdot t ) \cdot t +\frac12 a\cdot t^2 \\ & s &=& v_f\cdot t - a\cdot t \cdot t +\frac12 a\cdot t^2 \\ & s &=& v_f\cdot t - a\cdot t^2 +\frac12 a\cdot t^2 \\ & \mathbf{s} & \mathbf{=} & \mathbf{v_f\cdot t - \frac12 a\cdot t^2} \\ \end{array}\)

If 3n + 1 is a perfect square, show that n + 1 is the sum of three perfect squares.

3n + 1 is a perfect square:

\(\small{ \begin{array}{rcl} 3n+1 &=& a^2 \\ 3n &=& a^2 - 1 \\ n &=& \frac{a^2-1}{3}\\ \hline n+1 &=& \frac{a^2-1}{3} +1 \\ n+1 &=& \frac{a^2-1+3}{3}\\ n+1 &=& \frac{a^2+2}{3}\\ \end{array} }\)

Because \(\frac{a^2+2}{3}\) is a integer then \(a^2+2\) is divisible by 3, then \(a^2\) is not divisible by 3  and also \(a\) is not divisible by 3,

because if 3 is not a prime factor in \(a^2\) a perfect spuare, then 3 is not a prime factor in  \(a\)

Two numbers are not divisible by 3. It is  \(3b +1\) and \(3b + 2\)

1. We substitute \(a = 3b+1\)

\(\small{ \begin{array}{rcl} n+1 &=& \frac{a^2+2}{3}\qquad \text{substitute }\ a = 3b+1\\ n+1 &=& \frac{(3b+1)^2+2}{3} \\ n+1 &=& \frac{9b^2+6b+1+2}{3} \\ n+1 &=& \frac{9b^2+6b+3}{3} \\ n+1 &=& 3b^2+2b+1 \\ n+1 &=& b^2 + b^2 + b^2 +2b+1 \\ n+1 &=& b^2 + b^2 + (b+1)^2\\ \end{array} }\)

So n + 1 is the sum of three perfect squares and \(b = \frac{a-1}{3}\).

2. We substitute \(a = 3b+2\)

\(\small{ \begin{array}{rcl} n+1 &=& \frac{a^2+2}{3}\qquad \text{substitute }\ a = 3b+2\\ n+1 &=& \frac{(3b+2)^2+2}{3} \\ n+1 &=& \frac{9b^2+12b+4+2}{3} \\ n+1 &=& \frac{9b^2+12b+6}{3} \\ n+1 &=& 3b^2+4b+2 \\ n+1 &=& b^2 + b^2 + b^2 +2b+2b+1+1 \\ n+1 &=& b^2 + b^2+2b+1 + b^2 +2b+1 \\ n+1 &=& b^2 +(b+1)^2+(b+1)^2\\ \end{array} }\)

So n + 1 is the sum of three perfect squares and \(b = \frac{a-2}{3}\).

Example 1:

\(\small{ \begin{array}{rcl} a &=&7 \\ 3n+1 =a^2&=& 7^2 \qquad \rightarrow \qquad n=\frac{a^2-1}{3}=\frac{49-1}{3} = 16\\\\ n+1 &=& 16+1=17 \\ 17 &=& b^2+b^2+(b+1)^2 \qquad a=3b+1 \qquad \rightarrow b = \frac{a-1}{3} = \frac{7-1}{3} = 2\\ 17 &=& 2^2+2^2+3^2 = 4+4+9\\ \end{array} }\)

Example 2:

\(\small{ \begin{array}{rcl} a &=&8 \\ 3n+1 =a^2&=& 8^2 \qquad \rightarrow \qquad n=\frac{a^2-1}{3}=\frac{64-1}{3} = 21\\\\ n+1 &=& 21+1=22 \\ 22 &=& b^2+(b+1)^2+(b+1)^2 \qquad a=3b+2 \qquad \rightarrow b = \frac{a-2}{3} = \frac{8-2}{3} = 2\\ 22 &=& 2^2+3^2+3^2 = 4+9+9\\ \end{array} }\)

Assuming P = (x, 0) and Q = (0, y)

\(x^2+y^2=13^2\)

Differentiate with respect to time

\(2x\frac{dx}{dt}+2y\frac{dy}{dt}=0\)

Substitute 12 for dx/dt and for y and rearrange

\(\frac{dy}{dt}= -12\frac{x}{12}=-x\)

Use the first equation above to replace x

\(\frac{dy}{dt}=-\sqrt{13^2-12^2}=-5\)

So Q moves at 5 m/s down the y-axis.

Do you mean volume? Squares can't be possible like that.

Thank you heureka! That solves one of our question marks!

Hello.

Hello,

I'm trying to help my daughter with her math assignment, but this one I can't figure out. Would appreciate help, thanks!

Let f(x)= ax / (2x+3)

Investigate if you can define the 'a' so that f(f(x))=x

New edit, without mistake:

\(\begin{array}{rcl} a_{1} &=& x + (x+3) \\ \mathbf{a_{1}} & \mathbf{=} & \mathbf{2x+3} \\\\ a_{2} &=& x - (x+3) \\ \mathbf{a_{2}} & \mathbf{=} & \mathbf{ -3 } \end{array}\)

Sat 28/11/15

1) Music Report #10      Thanks Dbg4thebest       

   http://web2.0calc.com/questions/music-news-report-10

2) Social post:  It is like an great reunion! Me, TitaniumRome and Dragonlance

   http://web2.0calc.com/questions/8-1-7-how-do-i-put-this-in-the-calculator/new#edit

3) Basic fact wioth trigonometry - what does the co stand for in cosine?     Melody

    http://web2.0calc.com/questions/if-3n-1-is-a-perfect-square-show-that-n-1-is-the-sum-of-three-perfect-squares#r1

4) Statistics question from DragonSlayer554.    Thanks  Dragonlance.

    https://web2.0calc.com/questions/math-question-i-m-back

5) Calculus   Thanks Alan

   https://web2.0calc.com/questions/calculus-question_1#r2

6) Proof    Thanks Heureka

    http://web2.0calc.com/questions/if-3n-1-is-a-perfect-square-show-that-n-1-is-the-sum-of-three-perfect-squares#r1

7)  Intuitive problem.  Thanks guests, CPhill and Melody.

     http://web2.0calc.com/questions/please-give-an-intuitive-explanation#r6

8)   Proof    Thanks guys but I would really like another mathematician to look at this one. :/

     This was from last time but I would still like it to be discussed further :/

     https://web2.0calc.com/questions/help_67408

@@ What is Happening?  [Wrap4]   Fri 27/11/15   Sydney, Australia Time 3:35 pm   ♪ ♫

Hello everyone,

My weekend is already beginning but many of you will have to wait another day or so!  :)

We have had some great answers from CPhill, Anonymous4338, buubleman, Geno3141, Heureka, and Dancer04.  Thank you, you are all wonderful.     

Forum Improvement:

We now have an inbox for private messages.  This is great!    Thanks Mr Massow.   

Interest Posts:

If you ask or answer an interesting question, you can private message the address to me (with copy and paste) and I will include it.  Of course only members are able to do this.  I quite likely will not see it if you do not show me.  

1) Call for guests to become members.   Thanks Anonymous4338  :)

Hi guest,      

I appreciate that you offered an answer, I want everyone to participate and I hope that you are not offended by my involvement.  I also hope that you and the asker both learn someting positive from my input here.

In this case it was a necessary for you to state your assumption.

You see, this student, and many others, do not understand that they are solving an equation that must be equal to zero.

SO the most important thing was to teach the student why their question made no sense.

When answerers make assumptions they must state it.  

They may do this just by rewriting the question properly (as they assume it to be) 

OR they may make a song and dance about it like I have here.

BUT either way, the questioner must be told what question you are answering.   

-59992000000

I thought zoka123 would be looking for a general proof.    

Yes, this one looks reallly interesting.   Thanks Geno and Heureka.

I have put it aside for when I have more time. I want to look at both your answers  :)

Yes thank you anonymous4338 and guest,

General thank yous are nice but it is very appreciated if you say thank you on your own question or interest thread.

People like personal thank yous.  It is nice to say thank you, and ask for more clarification if neccessary, every time you recieve an answer.  :)

Thanks 4338 that is really sweet of you :)

Melody: Did you read my note in answer #4?. As I mentioned there, the only formula you need is the STANDRAD TVM formula for PV of a stream of payments into the future!!. It's listed there!. Look at my note in #4 above. The derivation of most of the standard TVM formulae was derived by Sir Isaac Newton some 300-350 years ago!. There are some modern formulas that are used in special but rare situations, which I have not come across yet.

NeedSupply: Most of the financial software that I have in my computer is proprietary, written and programmed into my computer by myself, because I worked as an Investment Banker for some 25 years. In addition to that, I have  a certain mathematical ability to fully understand the Math that is used in investments. In your case, because you are a student, I recommend that you buy youself a "Financial calculator". I would recommend that buy one that you can afford such as HP-10b11, on which you can do most, if not all, your problems that I have seen so far. I have not seen the price of it, but I believe you can buy one for as little as $30.00US. It has many memories as well as scientific functions, statistics, cash flow analysis, date and time calculations.....etc.

Good luck to you.

This is a mathematics site Mr Banker. Not a finanical site.

Why don't you show some ability and show the derivation of your formula.

I know the derivation has already been done by someone else so you should be able to find it easily enough on the net.

I have seen no evidence that you are capable of deriving it for yourself.

Okay, Guest.

2+3x/2=3x

First, (following PEMDAS), let's do 3x/2. The answer is 1.5x. Now our equation is 2+1.5x=3x.

Next, we can subtract 1.5x from each side. This gets us 2=1.5x.

Finally, divide each side by 1.5 to make sure we have a value of 1x on the right side. We now have 1 1/3=x

In conclusion, x= 1 1/3 (one and one-third).

Of course, for any viewers of this particular question, rest assured that 2+2 is, indeed, mathematically speaking, 4.

That's a hard one!

2+2 can be...

4

Fish

22

four

two plus two

2+2

twenty-two

This topic has been asked about more than once before, and I'm glad you brought it up again!

Thanks, Guest, for livening up our community!

Guest, please don't be rude!

Let me help you out here...

Lucas is 12 years old.

Isack is 3 years younger than Lucas since the equation is Lucas' age-3 years.

Since Lucas is 12, that means that Isack is 3 years younger than 12 years old, which means Isack is 9.

In conclusion, Isack is 9 years old!