Question

Help me to solve this question ​

Answer 1

$\Large{\underline{\underline{\mathfrak{\bf{Question}}}}}$

$\sf{\:If\:9^n+9^n+9^n\:=\:3^{2013}} \\ \sf{\:,what\:is\:the\:value\:of\:n}$

$\Large{\underline{\underline{\mathfrak{\bf{Solution}}}}}$

$\Large{\underline{\mathfrak{\bf{Given}}}}$

• $\sf{\:If\:9^n+9^n+9^n\:=\:3^{2013}}$

$\Large{\underline{\mathfrak{\bf{Find}}}}$

• Value of n

$\Large{\underline{\underline{\mathfrak{\bf{Explanation}}}}}$

We know,

$:\mapsto\sf{\red{\:If\:a^m\:+\:a^n\:=\:a^{m+n}}}$

So,

$\sf{\:9^n+9^n+9^n\:=\:3^{2013}} \\ \\ \\ \small\sf{\green{\:\:\:using\:above\:identity}} \\ \\ \\ :\mapsto\sf{\:9^{n+n+n}\:=\:3^{2013}} \\ \\ \\ :\mapsto\sf{\:9^{3n}\:=\:3^{2013}} \\ \\ \\ :\mapsto\sf{\:(3^2)^{3n}\:=\:3^{2013}} \\ \\ \\ \small\sf{\green{\:\:\:\:\bigstar\:(a^m)^n\:=\:a^{mn}}} \\ \\ \\ :\mapsto\sf{\:3^{6n}\:=\:3^{2013}} \\ \\ \\ \small\sf{\green{\:\:\:if\:a^m\:=\:a^n\:then\:\:m\:=\:m}} \\ \\ \\ :\mapsto\sf{\:6n\:=\:2013} \\ \\ \\ :\mapsto\sf{\red{\:n\:=\:\dfrac{2013}{6}}}$

$\Large{\underline{\mathfrak{\bf{Hence}}}}$

• Value of n = 2013/6

___________________________