Melody

@@ End of Day Wrap :     Sun 24/8/14    Sydney, Australia      Time 1:25am     (Really monday morning)   ♬  

Hello all,

Great weekend answers from xvxvxv, CPhill, AzizHusain, Alan, BlinxD, NinjaDevo and Will85237.  A big thank you to each of you.      

Edit: note to DS.  A bit thank you is a bit of a thankyou because each answerer gets a part of it.  Of course the thank you is infinite so the 'bit' is infinite as well.  Does that answer your question DS?

Now for some interest threads:

* 1) Why is  $$-3^2=-9\;\; not\;\; +9?$$       (Great answer from Ninja)

ok which one really is your question.

Technically irt should  be the first one but I suspect it was supposed to be the last one. I doubt that it is the middle one.  You needed more backets to make the meaning clear.  Which would our like solved?

$$\\\frac{5(2x+1)^{2}}{4}\times x+5 = 5x-1\\\\
\\\frac{5(2x+1)^{2}}{4}\times (x+5) = 5x-1\\\\
\frac{5(2x+1)^{2}}{4x+5} = 5x-1$$

When traveling downstream speed = boat + current = 20miles in 2 hours = 10miles/hour

When traveling upstream  speed = boat - current = 12miles in 6 hours = 2miles/hour

so we have 2 equations which must be solved simultaneously.

B+C=10

B-C=2

add and we get

2B=12

B=6

subtract and we get

2C=8

c=4

So the boat is traveling at 6 miles per hour and the current is traveling at 4 miles/hour.

$$$64x-\frac{4x}{3}$\\\\
$=\frac{64*3x}{3}-\frac{4x}{3}$\\\\
$=\frac{192x}{3}-\frac{4x}{3}$\\\\
$=\frac{188x}{3}$\\\\$$

This is not a question.

100*10^6 + 0.3*10^k = 1.3*10^8

$$\begin{array}{rll}
100*10^6 + 0.3*10^k &=& 1.3*10^8\\
10^8 + 0.3*10^k &=& 1.3*10^8\\
10^8-10^8 + 0.3*10^k &=& 1.3*10^8-10^8\\
0.3*10^k &=& 0.3*10^8\\
k &=& 8\\
\end{array}$$

Thanks for the question Takahiro 

I cheated (or used the resources available to me)

Thanks,

Flattery goes a long way with me  

$$\left( \frac{e^x-e^{-x}}{2}\right)^2-1\\\\
= \frac{(e^x-e^{-x})^2}{2^2}-1\\\\
= \frac{e^{2x}-2e^xe^{-x}+e^{-2x}}{4}-\frac{4}{4}\\\\
= \frac{e^{2x}-2+e^{-2x}}{4}-\frac{4}{4}\\\\
= \frac{e^{2x}+e^{-2x}-6}{4}\\\\
\mbox{Do you want me to keep going? It is probably finish now:)}\\\\
= \frac{e^{2x}+\frac{1}{e^{2x}}-6}{4}\\\\
= \frac{\frac{e^{4x}+1-6e^{2x}}{e^{2x}}}{4}\\\\
= \frac{e^{4x}+1-6e^{2x}}{4e^{2x}}\\\\
= \frac{e^{4x}-6e^{2x}+1}{4e^{2x}}\\\\$$

$$\frac{1}{\sqrt2}\\\\
=\frac{1\times \sqrt2}{\sqrt2\times \sqrt2}\\\\
=\frac{\sqrt2}{2}\\\\\\
or\;alternatively\\\\
\frac{\sqrt2}{2}\\\\
=\frac{\sqrt2\div \sqrt2}{2\div \sqrt2}\\\\
=\frac{1}{2^1\div 2^{1/2}}\\\\
=\frac{1}{2^{1-(1/2)}}\\\\
=\frac{1}{2^{(1/2)}}\\\\
=\frac{1}{\sqrt2 }\\\\$$