Melody

$${\mathtt{245\,678\,765\,432}}{\mathtt{\,-\,}}{\mathtt{23\,546\,432}} = {\mathtt{245\,655\,219\,000}}$$

245655219000+86x:4x

$${\frac{{\mathtt{245\,655\,219\,000}}}{{\mathtt{2}}}} = {\mathtt{122\,827\,609\,500}}$$

122827609500+43x:2x

And that is it!

Thank you rosala :-)

$$M^2+(M^2+4M)\\\\
=M^2+M^2+4M\\\\
=2M^2+4M$$

Why don't we do it on the site calculator and see what happens

A girl puts the number 6 on the display of her calculator, presses the square root button, multiplies the answer by 2 and then squares that answer.Her calculator would be showing:

This is without hitting the equal sign.

 sqrt 6 * 2 ^2 

$${\sqrt{{\mathtt{6}}}}{\mathtt{\,\times\,}}{{\mathtt{2}}}^{{\mathtt{2}}} = {\mathtt{9.797\: \!958\: \!971\: \!132\: \!712\: \!4}}$$

Below is the answer if you do hit the equal sign as you go through.

6 sqrt = * 2 = ^2

 $${\mathtt{sqrt6}}{\mathtt{\,\times\,}}{\mathtt{2}} = {\sqrt{{\mathtt{6}}}}{\mathtt{\,\times\,}}{\mathtt{2}}$$

$${\left({\mathtt{sqrt6}}{\mathtt{\,\times\,}}{\mathtt{2}}\right)}^{{\mathtt{2}}} = {{\left({\sqrt{{\mathtt{6}}}}{\mathtt{\,\times\,}}{\mathtt{2}}\right)}}^{{\mathtt{2}}}$$=  24

sqrt6 on this calculator is often entered as 6 sqrt on older calculators.

So it the equal sign is hit as you go through the question. Because it does so 

(1)multiply the answer

(2)square that answer

The only way you can get a running 'answer' like this is to keep hitting the equal sign.

Thank you Chris,  we do make a good pair of colourful eggs.   ♬  

I just gave you three more points too Chris.

I like being a colourful egg!   ♬  

Let x be the amount of water in tank A

Let m be the amount of water needed to fill tank A

So  the total capacity of tank A is (x+m)

----------------------

Let y be the amount of water in tank B

Let n be the amount of water needed to fill tank B

and the total capacity of tank B is (y+n)

-------------------------

We are being asked to find (n+m)

Now the capacity ratio of tank A to tank B is 7:3  so

$$\begin{array}{rll}
\frac{x+m}{y+n}&=&\frac{7}{3}\\\\
3(x+m)&=&7(y+n)\\\\
\end{array}$$

--------------------------

now there is x litres in tank A and m litres are needed to fill tank B so  

x-n=33 litres  ==>  x=33+n

Likewise

y-m=9 lites ==>  y=9+m

-----------------------------

$$\begin{array}{rll}
3(x+m)&=&7(y+n)\qquad \mbox{substituting for x and y we get;}\\
3(33+n+m)&=&7(9+m+n)\\
3*33+3(m+n)&=&7*9+7(m+n)\\
99+3(m+n)&=&63+7(m+n)\\
36&=&4(m+n)\\
9&=&m+n\\
\end{array}$$

so the amount of water to fill up both tanks completely is 9L

sin(2arctan(12/5))

I always do these by drawing the triangle

$$let\:\theta = atan\:\frac{12}{5}$$

sin(2arctan(12/5))

$$\\Sin(2 atan(12/5))\\
=sin(2\theta)\\
=2sin\theta cos\theta\\
=2\times \frac{12}{13}\times \frac{5}{13}\\
=\frac{120}{169}$$

(1 + 2 x^1/3  + x^2/3) 

This is as

$$1+2\sqrt[3]x+\sqrt[3]{x^2}\\\\
OR\\\\
1+2\sqrt[3]x+\left(\sqrt[3]x\right)^2\\\\$$

Thank you Rosala,

thank you for your kind words and your beauiful gifts.

Thank you for the beautiful flowers.  I was secretly hoping I would get some flowers off you soon.  You have made my day.  Thank you so much.    And, do you know what, purple is my favourite colour.   

                                                                       ♬ ♬ MELODY ♬ ♬