Advanced Exponent Properties (Powers of 2) | Algebra

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

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…

$2^1\ =\ 2\ \times\ 2^0\ =\ 2^0\ +\ 2^0$

$2^2\ =\ 2\ \times\ 2^1\ =\ 2^1\ +\ 2^1$

$2^3\ =\ 2\ \times\ 2^2\ =\ 2^2\ +\ 2^2$

$2^4\ =\ 2\ \times\ 2^3\ =\ 2^3\ +\ 2^3$

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 | $x$ |.