Does the product rule for exponents apply to a power of a power?
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)?
$\endgroup$ 24 Answers
$\begingroup$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.
$\endgroup$ 6 $\begingroup$It depends on how you write it.
$$3^{(4^5)}\ne 3^{(4^2)}.3^{(4^3)}$$
but
$$(3^4)^5=(3^4)^2.(3^4)^3$$
$\endgroup$ $\begingroup$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.
$\endgroup$ $\begingroup$A short answer is no. Using the product rule that works we have$$3^{4^2}3^{4^3} = 3^{4^2+4^3}\neq 3^{4^5} $$
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