Question

`Simplify: ( 5×(25)^(n+1) - 25 × (5)^(2n))/(5×(5)^(2n+3 )- (25)^(n+1​)) \\`

Correct answer please ​

Answer 1

`\green{\large\underline{\sf{Solution-}}}`

Given expression is

`\rm :\longmapsto\:\frac{5 * {25}^(n + 1) - 25 * {5}^(2n) }{5 * {5}^(2n + 3) - {25}^(n + 1) }`

can be rewritten as

`\rm \:  =  \: \frac{5 * { {(5}^(2) )}^(n + 1) - {5}^(2) * {5}^(2n) }{5 * {5}^(2n + 3) - {( {5}^(2) )}^(n + 1) }`

We know,

`\purple{\rm :\longmapsto\:\boxed{\tt{ {( {x}^(m) )}^(n) \: = \: {x}^(mn)}}} \\`

And

`\purple{\rm :\longmapsto\:\boxed{\tt{ \: \: {x}^(m) * {x}^(n) = {x}^(m + n) \: }}} \\`

So, using this identity, we

`\rm \:  =  \: \frac{5 * {5}^(2n + 2) - {5}^(2n + 2) }{{5}^(2n + 3 + 1) - {5}^(2n + 2) }`

`\rm \:  =  \: \frac{{5}^(2n + 2 + 1) - {5}^(2n + 2) }{{5}^(2n + 4) - {5}^(2n + 2) }`

can be further rewritten as

`\rm \:  =  \: \frac{{5}^(2n + 2 + 1) - {5}^(2n + 2) }{{5}^(2n + 2 + 2) - {5}^(2n + 2) }`

`\rm \:  =  \: \frac{ {5}^(2n + 2) (5 - 1)}{ {5}^(2n + 2) ( {5}^(2) - 1)}`

`\rm \:  =  \: (4)/(25 - 1)`

`\rm \:  =  \: (4)/(24)`

`\rm \:  =  \: (1)/(6)`

Hence,

`\rm :\longmapsto\:\boxed{\tt{ \frac{5 * {25}^(n + 1) - 25 * {5}^(2n) }{5 * {5}^(2n + 3) - {25}^(n + 1) } = (1)/(6) }}`