frasinscotland

ok thanks Hectitar, will give it a try.

Q   intersection   R  is { 11 }.                   Apologies,this site won't let me use intersection symbols.

 P    U    {11 }        is  { 8   12  13  15 }     U     {11}    which is

{ 8   11   12   13   15 }    ,your answer.

You can also,and probably should,use set algebra for this.  As below

P   U  (Q intersection R)       =  ( P U Q)    intersection    ( P  U R) .     You will get the same answer.

{ If anyone out there can tell me how to get maths symbols on this site,please do. This is getting quite frustrating. }   

The  ' h ' stands for ' hyperbolic'  .   The hyperbolic functions  sinh cosh tanh    are a completely different family of functions from sin cos tan  ( which are known as circular functions.)       Depending on how far you take maths,you may meet the hyperbolic functions later.  But first,get properly acquainted with the circular functions. That's fundamental if you are going to progress in calculus.

What you have written is the equation of a straight line     3y = 2x + 12.          Since there is an infinite number of points on a straight line ,there is an infinite solution set. You would need  another equation to obtain a specific solution. If you had another (linear) equation in  x  and  y  then your solution would be the unique point where the two lines intersected.

Don't get confused by the idea of   'per cent   X   another per cent'   which I think is bothering you. 'Per Cent' just means

'hundredth'.

So 40 per cent   X  70 per cent  is just   40 hundredths    X    70 hundredths.

Which is   40/100   X   70/100

Which is 4/10   X  7/10   =28/100     which is 28  hundredths   which is 28%. 

8^(-5)  / 8^3    = 1/8^5       X   1/8^3     add indices to get       1/8^8

or   8^(-5)  / 8^3   =  8^(-5)    X  8^(-3)       add indices   to get 8^(-8)    = 1/8^8,same thing.

Remember that  X^(-n)    = 1/ X^n
 

Company A : Principal sum  X (1.125)^5

Company B : Principlal sum X (1.102)^6

Notice that you don't need to actually involve the principal sum in the calculation.  You just need to evaluate the exponentials.

(1.125)^5      compared with  (1.102)^6      and see which is the least for the best deal.

differentiate the volume with respect to radius ie  dv/dr  of   4/3 (pi)(r^3)

= 4(pi)(r^2)   which is the formula for the surface area.

Well,you can use laws of indices to write this this as

(9/8)^5000   = (1.125)^5000  which is still a gigantic number.

If you now use the binomial theorem  and look at the first couple of terms you can see how quickly its series grows

We have ( 1 + x) ^n     = 1 +nx +[n(n-1)x^2]/2!    + [n(n-1)(n-2)x^3]/3! ...........

I'll just evaluate the first couple of terms to get

1  + 5000(0.125)  +[5000(4999)(0.125)^2]/2!       ......

= 1 + 625 + 195,273 ....

so by the time we only even get to the third term we are  in the hundreds of thousands. The 5000th term?  

Input (1.125)^5000 and see if you can make sense of the number!