What's the difference between $5^{2^4}$ and $\left(5^2\right)^4$?

Question

I wanted to know what exactly is the difference between these two expressions and why they return different results.

$$5^{2^4} \qquad\qquad\left(5^2\right)^4$$

The left one returns $152587890625$, while the right one gives $390625$.

I know the right one is called "powers of powers", but I don't know about the one on the left and how it works.

Answer 1

$$5^{2^4}=5^{(2^4)}=5^{16}$$

But

$$(5^2)^4=5^{(2\cdot 4)}=5^8$$

We have $$5^{16}=(5^8)^2$$

Answer 2

Be careful with parentheses. You might be used to the idea that $(5+2)+4 = 5+(2+4)$ and $(5 \times 2) \times 4 = 5 \times (2 \times 4)$, but the same doesn't stay true for powers! In particular, we don't have that $5^{(2^4)} = (5^2)^4$.

So, when writing $5 \times 2 \times 4$ it doesn't matter where the parentheses go, but it does for powers! In particular, the expression $5^{2^4}$ is interpreted at $5^{(2^4)}$, which is different from $(5^2)^4$. An easy way to see that they're different is just to evaluate them:

$$5^{(2^4)} = 5^{16} \qquad (5^2)^4 = 25^4 =5^8.$$

So now the question is - why do we assume $5^{2^4}$ means what it does? Well, notice how ${(a^b)}^c= a^{bc}$, so we already have a convenient notation for this. So, it's sensible to define the notation $a^{b^c}$ to refer to $a^{(b^c)}$.

Answer 3

This is effectively the same as Siong Thye Goh's answer, except with a little extra detail.

As others have remarked, the notation $5^{2^{4}}$ is really $5^{(2^{4})} = 5^{16}$. This is a simple application of powers.

On the other hand, by recalling what the power represents, we have $$ (5^{2})^{4} = 5^{2} \cdot 5^{2} \cdot 5^{2} \cdot 5^{2} = 5^{2+2+2+2} = 5^{4(2)} = 5^{8}. $$

Care and attention is required when dealing with powers of powers.