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Today, let’s discuss the relative placements of exponents on the number line.

We know what the graph of looks like:

It shows that when is positive, with increasing value of , increases very quickly (look at the first quadrant), but we don’t know exactly how it increases.

It also shows that when is negative, stays very close to 0. As decreases, the value of decreases by a very small amount.

Now note the spacing of the powers of 2 on the number line:

and so on…

So every power of 2 is equidistant from 0 and the next power. This means that a power of 2 would be much closer to 0 than the next higher powers. For example, is at the same distance from 0 as it is from .

But is much closer to 0 than it is to , etc.

Let’s look at a question based on this concept. Most people find it a bit tough if they do not understand this concept:

Question: Given that , which of the following values for yields the lowest value for ||?

A) 35

B) 90

C) 91

D) 95

E) 105

We need the lowest value of ||. We know that the smallest value any absolute value function can take is 0. So should be as close as possible to () to get the lowest value of ||.

Let’s try to simplify:

()

Which value should take such that is as close as possible to ?

is obviously larger than . But is it closer to or or higher powers of 2?

Let’s use the concept we have learned today – let’s compare with and .

So now if we compare these two with , we need to know whether is closer to 0 or closer to .

We already know that is equidistant from 0 and , so obviously it will be much closer to 0 than it will be to .

Hence, is much closer to than it is to or any other higher powers.

We should take the value 90 to minimize ||, therefore the answer is **(B)**.