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