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Simplify 32 + 5 / 6 + 23. We do have to pay attention to the order of operations, which
is designated by Please Excuse My Dear Aunt Sally - where the P stands for parentheses,
the E for exponents, then we have multiplication, division, addition and subtraction.
There aren't any parentheses, but actually that fraction bar separates the numerator
from the denominator, so it's like there's a set of parentheses around each of these.
Even if we could not see them, we must think of them as being there. So, let?s go ahead
and do the numerator, which is 32 + 5. Within that, we need to do the exponent first. So,
let's do a little scratch work over here on the side. 32 means 3 x 3, which is 9. So,
we have 9, and eventually we will worry about adding the 5. But let's not do that yet. Now
we will look at the denominator. And in the denominator, we do have an exponent, so that
has to be taken care of first. So let's think about that as well, over here in our scratch
area. 23 is equal to 2 x 2 x 2 which is 8. So in place of that 23, we will put an 8 and
of course 6 was supposed to be added to it.
Now we will continue to do just the top, just the bottom. So, 9 + 5 gives me a 14 and 6
+ 8 also gives me a 14. So, coincidentally since 14 goes into the top once and into the
bottom once, we get a result of 1 .
We have another problem to do. This one also has exponents and a big fraction bar in the
middle. So remembering Please Excuse My Dear Aunt Sally, we would do the parentheses first,
and remember that the fraction bar is like grouping the numerator pieces together and
the denominator, so we really do have a set of parentheses. Within those parentheses,
we do have exponents. So, since we have taken care of the parentheses, now we?ve got to
deal with exponents. I am going to do my scratch work over here. 25 is equal to 2 x 2 x 2 x
2 x 2. I think an easy way to look at that would be to group them a little bit. This
is 4 and 2 x 2 x 2 is 8. 4 x 8 is equal to 32. So instead of the 25, I will put in its
value which is 32. I also have a 52, which we all know to be 5 x 5 which is 25. So we
will put the subtraction sign and our 25. Now in the denominator of the original problem,
I have 33. So thinking about that, over here to the side, because I don?t want to mess
up the problem, I want to do all my messy stuff on the right, I am going to have 3 x
3 x 3, which turns out to be 9 x 3, or 27. So down here, I?ve got 27 and then I?m going
to add 8. Now remember that the fraction bar divides the numerator from the denominator
and it?s like having a set of parentheses around the top and around the bottom. So 32
-- 25. I?ll do some more messy work over here on the side. 32 -- 25, if I do that
by hand, I discover that the answer is 7. So, the simplified version of the numerator
is 7. Now I will add 27 + 8 getting 35. Now I could say that I?m done, except that this
fraction is not in its simplest form. I can take 7 into the top, the numerator, and I
can take 7 into the bottom, the denominator. 7 goes into the denominator 5 times and I
end up with 1/5 as the answer.
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