Step-by-step explanation:
[tex] \frac{1}{ {a}^{n} } = {a}^{ - n} [/tex]
~Math Induction
[tex] \: [/tex]
—Prove that n = 1
[tex] \frac{1}{ {a}^{1} } = {a}^{ - 1} [/tex]
[tex] \frac{1}{a} = \frac{1}{a} \to \sf proved[/tex]
[tex] \: [/tex]
—Prove that n = k
[tex] \frac{1}{{a}^{n}} = {a}^{n} [/tex]
[tex] \frac{1}{{a}^{k}} = {a}^{-k} , k \in \mathbb{R}[/tex]
[tex] \: [/tex]
—Prove that n = k + 1
[tex] \frac{1}{ {a}^{k + 1} } = {a}^{ - (k + 1)} [/tex]
[tex] \frac{1}{ {a}^{k + 1} } = {a}^{ - k} \times {a}^{ - 1} [/tex]
[tex] \frac{1}{ {a}^{k + 1} } = \frac{1}{ {a}^{k} } \times \frac{1}{ {a}^{1} } [/tex]
[tex] \frac{1}{ {a}^{k + 1} } = \frac{1}{ {a}^{k + 1} } \: \: \rm proved[/tex]
[tex] \: [/tex]
Proved that 1/a^n = a^-n
