Intereting Posts

Number fields with all degrees equal to a power of three
Pair of straight lines
$\lim\limits_{n\to\infty} \frac{n}{\sqrt{n!}} =e$
Cardinality of a set of closed intervals
Limit of the geometric sequence
How to calculate the following limit: $\lim_{x\to+\infty} \sqrt{n}(\sqrt{x}-1)$?
Show that $\rm lcm(a,b)=ab \iff gcd(a,b)=1$
Are open sets in $\mathbb{R}^2$ countable unions of disjoint open rectangles?
What does it mean for something to be true but not provable in peano arithmetic?
Why is the number of subgroups of a finite group G of order a fixed p-power congruent to 1 modulo p?
Understanding isomorphic equivalences of tensor product
Will $2$ linear equations with $2$ unknowns always have a solution?
$S := \{x \in \Bbb R^3: ||x||_2 = 1 \}$ and $T: S^2 \to \Bbb R$ is a continuous function. Is $T$ injective?
Proof of Nesbitt's Inequality: $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\ge \frac{3}{2}$?
When do Pell equation results imply applicability of the “Vieta jumping”-method to a given conic?

I’m so puzzled about this:

$$a^{b^c} = a^{(b^c)} \neq (a^b)^c = a^{(bc)}.$$

Why isn’t $a^{b^c}$ equal to $a^{(bc)}$? Why is $a^{b^c}$ instead equal to $a^{(b^c)}$? And how is it possible that $(a^b)^c = a^{(bc)}$?

- Biggest powers NOT containing all digits.
- Which step in this process allows me to erroneously conclude that $i = 1$
- Non-integer powers of negative numbers
- Given $a>b>2$ both positive integers, which of $a^b$ and $b^a$ is larger?
- Confusion regarding taking the square root given an absolute value condition.
- For which complex $a,\,b,\,c$ does $(a^b)^c=a^{bc}$ hold?

My mind is pretty much exploding from trying to understand this.

- Why do some people place the differential at the beginning of their integral?
- Generalized power rule for derivatives
- What are the common abbreviation for minimum in equations?
- Indexed Family of Sets
- A basic question about exponentiation
- Matrix raised to a matrix
- Why is $x^0 = 1$ except when $x = 0$?
- Modular Arithmetic with Powers and Large Numbers
- How to find the last digit of $3^{1000}$?
- Is the equation $\phi(\pi(\phi^\pi)) = 1$ true? And if so, how?

That $a^{b^c}$ stands for $a^{(b^c)}$ rather than for $(a^b)^c$ is merely a notational convention; we say that the exponentiation notation associates to the right (whereas arithmetic operations associate to the left, so that $a-b-c$ means $(a-b)-c$ rather than $a-(b-c)$).

The fact that $(a^b)^c=a^{b\times c}$ is easy to understand: $a^b$ is obtained by multiplying together a sequence of $b$ copies of $a$, and $(a^b)^c$ is obtained by multiplying together $c$ such products; writing all this out in terms of copies of $a$ means that $b\times c$ such copies have been multiplied together. Now you can see also why this is not equal to $a^{(b^c)}$ which is obtained by multiplying together $b^c$ copies of $a$.

Finally the fact that $(a^b)^c=a^{b\times c}$ explains why the convention is that exponentiation notation associates to the right: both $a^{(b^c)}$ and $(a^b)^c$ are useful expressions, but since the latter can be more easily written as $a^{bc}$, one might as well reserve $a^{b^c}$ to stand for the former, which has no such easy alternative. It might seem that $a^{b^c}$ is such an enormous product that is unlikely to be useful; it is however encountered surprisingly often in some contexts.

Note that $(2^2)^3=4^3=64$, whereas $2^{(2^3)}=2^8=256$ (and incidentally $2^{2\cdot 3}=2^6=64$).

First question,

Why isn’t $a^{b^c}$ equal to $a^{bc}$

assuming the remaining questions already answered. It is a matter of convention. Since $a^{(b^c)} \ne (a^b)^c$ in general, when we write $a^{b^c}$ we need convention to say which of these two we mean. Since $(a^b)^c = a^{bc}$, there is already a short way to write that, so we use $a^{b^c}$ for the other one.

First concept: *Multiplication is NOT the same as exponentiation.*

The question you are asking is, basically, why is $3^2 $ not the same as $3 (2)$ or $3 \times 2$?

Now, if we just take a look at an example problem involving your question:$$3^{2^{3}} = 3^{8} \ne {\left(3^{2}\right)^3}\ne 3^{2\times 3}\ne3^6$$

Notice that $3^{2^3}=3^8$ since $2^3 = 8$.

You are asking why multiplication is not the same as exponentiation, essentially. My answer is that multiplication is pertaining to repeated addition whereas exponentiation is pertaining to repeated multiplication.

If we look at fundamentals, $(a^b)^c$ is multiplying $a^b$ to itself $c$ times. Now if we multiply $a^b$ to itself an arbitrary number of times, then we are adding the power (here, $b$) to itself the same number of times. If you add something to itself some number of times, then you just multiply it by the number of times you are adding something to itself. Example: $$(2^3)^2 = 2^3 \times 2^3 = 2^{3 + 3} = 2^6 = 2^{3 \cdot 2}$$

And… $a^{b^c}$ means that you are multiplying $b$ to itself $c$ number of times.

**For casual readers: skip to this part.**

In $(a^b)^c$, we multiply $b$ to $c$… and in $a^{b^c}$, we raise $b$ to the power $c$. As I have clearly repeated, exponentiation is not the same as addition is not the same as exponentiation. In other words, $bc \ne b^c \ne b + c$ always.

Without parentheses: addition, subtraction, multiplication, and division should be evaluated **from left to right**. Like: $1 + 3 + 5 = 4 + 5 = 9$.

This rule **does NOT** apply for exponentiation, without parentheses, for exponentiation, we’ll go from **RIGHT to LEFT**.

For example, to evaluate: $2^{2^3}$, we must evaluate $2^3 = 8$ first, so: $2^{2^3} = 2^8 = 256$.

So, without any parentheses, $a^{b^c}$ is the same as $a^{\left(b^c\right)}$, since we must go from **RIGHT to LEFT**.

For your second problem, why ${(a^b)}^c = a^{b.c}$.

If we take the sum of some number $a$ for $n$ times, we’ll have **multiplication**, i.e $a \times n = \underbrace{a + a + a + … + a}_{n \mbox { times}}$.

If we multiply some number $a$ for $n$ times, we’ll have **exponentiation**, i.e $a ^ n = \underbrace{a \times a \times a \times … \times a}_{n \mbox { times}}$.

- $2 \times 3 = 2 + 2 + 2 = 6$
- $2 ^ 3 = 2 \times 2 \times 2 = 4 \times 2 = 8$
- $3 \times 4 = 3 + 3 + 3 + 3 = 12$
- $3 ^ 4 = 3 \times 3 \times 3 \times 3 = 9 \times 3 \times 3 = 27 \times 3 = 81$

- $a ^ m \times a ^ n = a^{m+n}$

## Proof

$a ^ m \times a ^ n = \underbrace{\underbrace{a \times a \times a \times … \times a}_{m \mbox { times}} \times \underbrace{a \times a \times a \times … \times a}_{n \mbox { times}}}_{m + n \mbox { times}} = a^{m+n}$.

It’s like 2 apples together with 3 apples becomes 2 + 3 = 5 apples. You have $n$ copies of $a$ together with another $m$ copies of $a$, you’ll get $m + n$ copies of $a$.

- $(a ^ m)^n = a^{m.n}$

## Proof

$(a^m)^n$ basically means that you take the result of $a^m$, then raise the whole stuff to the power of $n$, or in other words, multiply $n$ copies of it together.

$(a^m)^n = \underbrace{a^m \times a^m \times … \times a^m}_{n \mbox{ times}}$

Now, think of 5 groups of apples, such that that each group has exactly 2 apples. So there’ll be a total of 2 x 5 = 10 apples. Each $a^m$ has $m$ copies of $a$, and there are $n$ copies of $a^m$, or in other words, there are $n$ groups, in which each group has $m$ copies of $a$. So there’ll be a total of $m \times n$ copies of $a$. So:

$(a^m)^n = \underbrace{\underbrace{a \times a \times … \times a}_{m \mbox{ times}} \times \underbrace{a \times a \times … \times a}_{m \mbox{ times}} \times … \times \underbrace{a \times a \times … \times a}_{m \mbox{ times}}}_{n \mbox { times}} = a^{m.n}$.

- Is there a coherent sheaf which is not a quotient of locally free sheaf?
- How do the prime ideals of $\mathbb{Z}_{k}$ look like?
- Differentiating a function of a variable with respect to the variable's derivative
- If the expectation $\langle v,Mv \rangle$ of an operator is $0$ for all $v$ is the operator $0$?
- Hermitian matrix such that $4M^5+2M^3+M=7I_n$
- How to determine $\Omega(T)$?
- The homomorphism defined by the system of genus characters
- $O(n)$ algorithm to determine a number that appears more than $n/2$ times in an array of size $n$
- Coupon Problem generalized, or Birthday problem backward.
- A property of positive definite matrices
- Value of cyclotomic polynomial evaluated at 1
- finding inverse of $x\bmod y$
- What are some algebraically closed fields?
- Prove that $\lim f(x) =0$ and $\lim (f(2x)-f(x))/x =0$ imply $\lim f(x)/x =0$
- $|f(x)-f(y)|\le(x-y)^2$ without gaplessness