সূচক সংক্রান্ত সমীকরণ ও অভেদ [Equations and Identities Involving Indices]
1) যখন a, m, n বাস্তব সংখ্যা [tex]{a^m} = {a^n}[/tex] হলে [tex]m = n[/tex] যখন [tex]a \ne 0, \pm 1[/tex]
[tex]\begin{array}{l}
{a^m} = {a^n}\\
\Rightarrow \frac{{{a^m}}}{{{a^n}}} = 1\\
\Rightarrow {a^{m - n}} = {a^0}\left[ {a \ne 0, \pm 1} \right]\\
\Rightarrow m - n = 0\\
\Rightarrow m = n
\end{array}[/tex]
2) a, b, m বাস্তব সংখ্যা এবং [tex]{a^m} = {b^m}[/tex] হলে a = b যখন [tex]m \ne 0[/tex]
[tex]{a^m} = {b^m}[/tex] এর উভয়দিকে [tex]{b^m}[/tex] দিয়ে ভাগ করে পাই ।
[tex]\begin{array}{l}
{a^m} = {b^m}\\
\Rightarrow \frac{{{a^m}}}{{{b^m}}} = 1\\
\Rightarrow {\left( {\frac{a}{b}} \right)^m} = 1\\
\Rightarrow \frac{a}{b} = 1\left[ {m \ne 0} \right]\\
\Rightarrow a = b
\end{array}[/tex]
