Melody

@@ End of Day Wrap :    Fri 1/8/14     Sydney, Australia     Time 10:35am        ♬

Hello everyone,        ♫♪

Today our wonderful answerers were Alan, DragonSlayer554, NinjaDevo, Will8523, Rosala, CPhill, Sasini and AzizHusain.  Thank you.                         ♪ ♫ 

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 Some of these would make really nice additions to your 'Great questions to learn from' thread Ninja. 

I'll try and categorise them for you.   

 Working with logatithms. Hard one.                  ♪ ♫ 

Yes I can do that Stu.  It is quite nifty and little used.

$$7^4 = 49^2=(50-1)^2=2500-100+1=2401$$

Thinks about perfect squares.

$$(a-b)^2=a^2-2ab+b^2$$

You can prove this easily enough by doing the expansion but you should just know it anyway.  So

$$7^4\\
=(7^2)^2\\
=49^2\\
=(50-1)^2\\
=50^2-2*1*50+1^2\\
=2500-100+1\\
=2401$$

There are no negative numbers here Stu. 

7^765 * 7^450 * 7^-822 * 7^-389.

$$7^{765} * 7^{450} * 7^{-822} * 7^{-389}\\\
=7^{765+450-822-389}\\\\$$

$${\mathtt{765}}{\mathtt{\,\small\textbf+\,}}{\mathtt{450}}{\mathtt{\,-\,}}{\mathtt{822}}{\mathtt{\,-\,}}{\mathtt{389}} = {\mathtt{4}}$$

$$7^4 = 49^2=(50-1)^2=2500-100+1=2401$$     

I've done a little more working than necessary because I didn't feel like using a calculator except I was lazy with the addition.

Two excellent answers - Thank you.      

Very pretty Ninja  

This question is incomplete.

This question is not complete.  

I always like to draw a triangle to answer these.