1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ $5^{2^4}$ represents $5^{\left(2^4\right)}$ $\endgroup$ – Henry May 13 '18 at 22:50
2
$\begingroup$

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

But

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

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

$\endgroup$
1
$\begingroup$

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 $a^{(b^c)}$.

$\endgroup$
0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.