AMC 10 Daily Practice - Exponent

Factorizing the numerator, then becomes which is equal to which is .

∵ ,

∴ ,

∴ ①.

∵ ,

∴ ,

∴ ②.

Solve the system of equations ① and ②: , we get .

We write . Hence we see that our number is a weighted average of and , extremely heavily weighted toward . Hence the number is ever so slightly less than , so the answer is .

Let's set this value equal to . We can write . Multiplying by on both sides, we get . Now let's take a look at the answer choices. We notice that , choice , can be written as . Plugging this into out equation above, we get . The right side is larger than the left side because . This means that our original value, , must be less than . The only answer that is less than is so our answer is .

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We can ignore the 's on the end because they won't really affect the fraction. So, the answer is very very very close but less than the new fraction.

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So, our final answer is very close but not quite , and therefore the greatest integer less than the number is .

Let and . Then our fraction can be written as . Notice that . So, . And our only answer choice less than is .

Let . Multiply both sides by , and expand. Rearranging the terms, we get . The left side is strictly decreasing, and it is negative when . This means that the answer must be less than ; therefore the answer is .

A faster solution. Recognize that for exponents of this size will be enormously greater than , so the terms involving will actually have very little effect on the quotient. Now we know the answer will be very close to .

Notice that the terms being added on to the top and bottom are in the ratio with each other, so they must pull the ratio down from very slightly. (In the same way that a new test score lower than your current cumulative grade always must pull that grade downward.) Answer: .

We can compare the given value to each of our answer choices. We already know that it is greater than because otherwise there would have been a smaller answer, so we move onto . We get:.

Cross multiply to get: .

Cancel out and divide by to get . We know that , which means the expression is less than so the answer is .

Notice how can be rewritten as . Note that , so the greatest integer less than or equal to is or .

For positive , if then . Let .

Then . So answer is less than , which leaves only one choice, .

Try long division, and notice putting as the denominator is too big and putting is too small. So we know that the answer is between and , yielding as our answer.

We know this will be between and because and . is the only option choice in this range.