Question

Find the value of (i)^(496)^397 . "i" is iota.​

Answer 1

ANSWER:

To find:

• $\displaystyle (i)^{(496)^{(397)}}$

Solution:

We are given that,

$\displaystyle\implies (i)^{(496)^{(397)}}$

We know that,

$\displaystyle \hookrightarrow a^{x^y}=a^{xy}$

So,

$\displaystyle \implies (i)^{(496)\times(397)}$

Now, this question may seem difficult but it is not.

We need not multiply the exponents, 496 & 397, to solve it, but make it easy to solve.

So,

We know that, 4 is a factor of 496.

$\displaystyle (\implies 4 \times 124 = 496)$

So,

$\displaystyle \implies (i)^{(496\times397)}$

$\displaystyle \implies (i)^{(4\times124\times397)}$

Now,let us assume that,

$\displaystyle \hookrightarrow 124\times397 = x$

So,

$\displaystyle \implies (i)^{4\times x}$

$\displaystyle \implies (i)^{4x}$

$\displaystyle \implies (i)^{4^x}$

We know that,

$\displaystyle \hookrightarrow i^4=1$

So,

$\displaystyle \implies (i)^{4^x}$

$\displaystyle \implies 1^x$

Substituting value of x,

$\displaystyle \implies 1^{( 124\times397 )}$

Hence,

$\displaystyle \implies 1$

Therefore,

$\displaystyle \implies\bf (i)^{(496)^{(397)}}=1$