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Algebra - Advanced Exponent Properties (Powers of 2)

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

 

We know what the graph of 2^x looks like:

 

Graph of 2^x

 

It shows that when x is positive, with increasing value of x, 2^x increases very quickly (look at the first quadrant), but we don’t know exactly how it increases.

It also shows that when x is negative, 2^x stays very close to 0. As x decreases, the value of 2^x decreases by a very small amount.

 

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

Number line jpg

 

2^0 = 1

2^1 = 2

2^2 = 4

2^3 = 8

and so on…

 

2^1 = 2 * 2^0 = 2^0 + 2^0

2^2 = 2 * 2^1 = 2^1 + 2^1

2^3 = 2 * 2^2 = 2^2 + 2^2

2^4 = 2 * 2^3 = 2^3 + 2^3

 

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, 2^2 is at the same distance from 0 as it is from 2^3.

But 2^2 is much closer to 0 than it is to 2^4, 2^5 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 x = 2^b – (8^30 + 16^5), which of the following values for b yields the lowest value for |x|?

A) 35

B) 90

C) 91

D) 95

E) 105

 

We need the lowest value of |x|. We know that the smallest value any absolute value function can take is 0. So 2^b should be as close as possible to (8^30 + 16^5) to get the lowest value of |x|.

 

Let’s try to simplify:

(8^30 + 16^5)

= (2^3)^30 + (2^4)^5

= 2^90 + 2^20

 

Which value should b take such that 2^b is as close as possible to 2^90 + 2^20?

2^90 + 2^20 is obviously larger than 2^90. But is it closer to 2^90 or 2^91 or higher powers of 2?

 

Let’s use the concept we have learned today – let’s compare 2^90 + 2^20 with 2^90 and 2^91.

2^90 = 2^90 + 0

2^91 = 2^90 + 2^90

 

So now if we compare these two with 2^90 + 2^20, we need to know whether 2^20 is closer to 0 or closer to 2^90.

 

We already know that 2^20 is equidistant from 0 and 2^21, so obviously it will be much closer to 0 than it will be to 2^90.

Hence, 2^90 + 2^20 is much closer to 2^90 than it is to 2^91 or any other higher powers.

We should take the value 90 to minimize |x|, therefore the answer is B.

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