Question

if (m)ⁿ = 64,when m and n are positive integers,then find the value of (n)^mn

Answer 1

$$\begin{gathered} (m{)}^n=64 \\ Nowitisgiventhatmandnarepositiveintegerswhichmeansnmaybe1,2,3,...... \\ Henceputtingn=1,weget(m{)}^1=64\Rightarrow m=64 \\ \therefore (n{)}^{mn}=(1{)}^{64}=1 \\ Ifn=2then(m{)}^2=64\Rightarrow m=\sqrt{64}\Rightarrow m=8 \\ \therefore (n{)}^{mn}=(2{)}^{8\times 2}=(2{)}^{16}=65536 \\ Ifn=3then(m{)}^3=64\Rightarrow m=\sqrt[3]{64}\Rightarrow m=4 \\ \therefore (n{)}^{mn}=(3{)}^{4\times 3}=(3{)}^{12}=531441 \\ Ifn=4then(m{)}^4=64\Rightarrow misnotaninteger \\ Ifn=5then(m{)}^5=64\Rightarrow misnotaninteger \\ Ifn=6then(m{)}^6=64\Rightarrow m=\sqrt[6]{64}\Rightarrow m=2 \\ \therefore (n{)}^{mn}=(6{)}^{2\times 6}=(6{)}^{12} \\ Andforn=7onwards,mwillnotbeaninteger \end{gathered}$$