Question

Guys please answer this. Im in dire need of your help! ):

why can we write 1/a^n=a^-n?

Answer 1

Explanation:

`\frac{1}{ {a}^(n) } = {a}^( - n)`

~Math Induction

`\:`

—Prove that n = 1

`\frac{1}{ {a}^(1) } = {a}^( - 1)`

`(1)/(a) = (1)/(a) \to \sf proved`

`\:`

—Prove that n = k

`\frac{1}{{a}^(n)} = {a}^(n)`

`\frac{1}{{a}^(k)} = {a}^(-k) , k \in \mathbb{R}`

`\:`

—Prove that n = k + 1

`\frac{1}{ {a}^(k + 1) } = {a}^( - (k + 1))`

`\frac{1}{ {a}^(k + 1) } = {a}^( - k) * {a}^( - 1)`

`\frac{1}{ {a}^(k + 1) } = \frac{1}{ {a}^(k) } * \frac{1}{ {a}^(1) }`

`\frac{1}{ {a}^(k + 1) } = \frac{1}{ {a}^(k + 1) } \: \: \rm proved`

`\:`

Proved that 1/a^n = a^-n

Answer 2

Try to understand by substitute a number in a and n.

Explanation:

For example,

a=2, n=1

a^n= 2^1= 2

`(1)/(2)` = 0.5

2^(-1)=0.5