Question number 27. The value of the expression given here, which I will not try to pronounce for you, is equal to which of the following? So let me first begin by splitting this fraction into a sum of two fractions where I will recover the underlying portions here. So let me copy those. That will be one term and I will write the exponents as powers of 2. So I have 2 to the power 2 to the power 12 and we notice that that's the pattern in general. Here I have exponents 2 to the power 12, here 2 to the power 11, 2 to the power 10, and so on. But I'm splitting off the last term and then it has to have a denominator here of 3 and you can see that the exponent is a power of 2 and that power is exactly 11 and then plus the remaining terms. So the denominator has to be the same here, 3 to the power 2 to the power 11, and then we have our first factor here, which I will just write as a 5 and the remaining factors I will write as powers of 2 to a power of 2. So this is first 2 to the power 2 to the first power, 3 to the power 2 to the first, and then we continue like that until we get to 2 to the power 2 to the power 12 plus 3 to the power 2 to the power 12, like so. And then I'll rewrite this sum in the following way. I have my two factors, let me just copy those, and then my 5 and I'll use the product notation here where each factor will look like a power of 2 to a power of 2 with the denominator here, 3 to the power 2 to a power, and the index runs from 1 all the way up to 11. So here we have 11 terms, but we notice that the denominator here is too large. When you multiply all of these out. Why is that? Well, we have 2 to the power p and the exponent, and we're adding essentially 11 terms like that, so p runs from 1 to 11, so this is the exponent of 3. And using the geometric summation formula, we come up with 2 to the power 12 minus 2 over here. So what we need to do is we need to compensate and multiply by a compensating factor, so we need to multiply by the factor 3 to the power 2 to the power 11 divided by 9, like that. And that, the 9 here is 3 squared, so that compensates for this 2, and then 3 to the power 2 to the 11th times itself is 3 to the power 2 to the 12th, so that's how we build that factor. And then we can simplify this, so let me write this again. So we now have 2 to the 2 to the 12th divided by 3 to the 2 to the 11th plus 5 to the 9th, and then 3 to the 2 to the 11th, and then we have our product here with 11 factors, like that, so that's equal to the original expression. And here we simplify, we use the pattern that looks like this, 1 plus a, and then the exponent would be 2 if b is equal to 1, 1 plus a to the 4th if b is equal to 2, 1 plus a to the 8th if b is equal to 3, like that, and this comes out to a geometric series, like that, where a is equal to 4 9ths, and p is equal to 1, 2, all the way up to 11, except that 11 terms will give us 2 to the 11th terms, or I should say, to be more correct, 11 factors in the product will give us 2 to the 11th terms in the sum. So we have a geometric sum like that, so this comes out to the sum from p running from 0 to 2 to the 11th minus 1, and then we have 4 9ths to the power p. And that sum we then compute, that's 9 5ths, and then we have 1 minus 4 9ths to the power 2 to the power 11. And those are all the ingredients that we need, so let me write down here the final expression that will give us the answer. So finally we have 2 to the 2 to the 12th divided by 3 to the 2 to the 11th, we have our next term, 5 9ths times 3 to the 2 to the 11th, and then we have times 9 5ths, so we see there will be some cancellation already, and then 1 minus, let me write this as 4 9ths here to the power 2 to the power 11, like that, and then let's discuss what cancels with what. So we have a cancellation here, obviously the 5 9ths and the 9 5ths will cancel, like so, and then carefully after distributing the exponents, exactly these terms here cancel, and what remains is the 3 to the power 2 to the power 11th. And since we see here that 2 to the power 11th is 2048, that's exactly what we're looking at. 3 to the power 2048 is the only, is the end result of this long computation, and so we choose that as our answer, and the answer then comes out to C.

mathematical expression

geometric series

geometric summation

powers of three

simplification