How are the power rules for quotients and products similar? How are they different? Refer specifically to the way that the rule
1 answer:
Power rules for both quotient and product sums are useful to simplify large exponential form (of the same base)
The difference is in the rule. For quotient sum, the powers are subtracted, while for product sum, the powers are added up.
An example for quotient sum
Using the principle of simplifying fractions, we can cancel out ten 7s from both numerator and denominator, leaving us with only three 7s on the numerator which gives us
. This working out could be simplified by doing 
An example for product sum
. There is a total of eleven 9s if we were to work out the product sum the long way. This could be simplified by doing 
The difference is in the rule. For quotient sum, the powers are subtracted, while for product sum, the powers are added up.
An example for quotient sum
Using the principle of simplifying fractions, we can cancel out ten 7s from both numerator and denominator, leaving us with only three 7s on the numerator which gives us
An example for product sum
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Answer:
volume V of the solid
Step-by-step explanation:
The situation is depicted in the picture attached
(see picture)
First, we divide the segment [0, 5] on the X-axis into n equal parts of length 5/n each
[0, 5/n], [5/n, 2(5/n)], [2(5/n), 3(5/n)],..., [(n-1)(5/n), 5]
Now, we slice our solid into n slices.
Each slice is a quarter of cylinder 5/n thick and has a radius of
-k(5/n) + 5 for each k = 1,2,..., n (see picture)
So the volume of each slice is
for k=1,2,..., n
We then add up the volumes of all these slices
Notice that the last term of the sum vanishes. After making up the expression a little, we get
But
we also know that
and
so we have, after replacing and simplifying, the sum of the slices equals
Now we take the limit when n tends to infinite (the slices get thinner and thinner)
and the volume V of our solid is
