\(2^{2^3}\) is equivalent to \(2^{(2^3)}\), which is equivalent to \(2^8\). Similarly, \(2^{3^2}\) is equivalent to \(2^9\).
The positive difference can be calculated several ways.
Without a calculator and with some manipulations
$$ 2^9-2^8 $$
$$ =2^8(2^1-2^0) $$
$$=2^8(2-1)$$
$$=2^8$$
$$\boxed{=256} $$
Find \(2^9\) and \(2^8\) in our head, or with a calculator
$$2^9 = 512 $$
$$2^8 = 256 $$
$$2^9-2^8 $$
$$=512-256 $$
$$=\boxed{256}$$
It may be beneficial to use mental math and practice calculating the powers of 2. It has many practical applications (binary code) and is common in mathematics. Use your fingers to keep track of which power your at, and double the numbers in your head until it gets too difficult. You may be able to get to \(2^9\) without any trouble.
$$2^0=1$$
$$2^1=2$$
$$2^2=4$$
$$2^3=8$$
$$2^n=...$$
What is the positive difference of the two following expressions?
$$ 2^{2^3} \hspace{15px} 2^{3^2} $$