Does the product rule for exponents apply to a power of a power?

Question

Does the product rule for exponents apply to a power of a power? For example: If I have 3^(4^5), will that be equal to 3^(4^2)*3^(4^3)?

Answer 1

First, note that exponents are evaluated right-to-left. What I mean by this is that $$ {a^b}^c $$ by convention means $a^{(b^c)}$. It is important to have such a convention, since $$ a^{(b^c)} \neq (a^b)^c $$ in general. For example, $$ 2^{(3^4)} \neq (2^3)^4 \, . $$ Because of this, your question as it stands is a little ambiguous. When you write 'a^b^c', do you mean $${(a^b)}^c$$ or $$a^{(b^c)} \, ?$$ If you mean the former, then the product rule for exponents does hold: $$ (a^b)^c \times (a^b)^d = (a^b)^{c+d} \, . $$ To explain why, try setting $k=a^b$. Then we have $$ k^c \times k^d = k^{c+d} \, , $$ which is the familiar product formula. However, if by a^b^c you meant $$ a^{(b^c)} \, , $$ then the product formula does not apply in the way you suggested. Instead, $$ a^{b^c} \times a^{b^d} = a^{b^c+b^d} \, . $$ Try working out why this is the case, and ask me if you have any questions.

Answer 2

It applies in these terms: $$3^{4^2}\times 3^{4^3}=3^{4^2+4^3}\stackrel{\text{assuming you like this}}=3^{4^2(4^{3-2}+1)}=3^{5\cdot4^2}$$

And I don't think you want claim that $3^{4^5}=3^{80}$. I sure don't.

Answer 3

It depends on how you write it.

$$3^{(4^5)}\ne 3^{(4^2)}.3^{(4^3)}$$

but

$$(3^4)^5=(3^4)^2.(3^4)^3$$

Answer 4

A short answer is no. Using the product rule that works we have $$3^{4^2}3^{4^3} = 3^{4^2+4^3}\neq 3^{4^5} $$