Question

Can someone explain this to me please?

Answer 1

123

Explanation:

Squaring the given equation gives

`t^2+(2\cdot t \cdot (1)/(t) )+ (1)/(t^2)=9.`

The
`t` and
`(1)/(t)` terms cancel out nicely, which is one of the reasons for squaring the equation.

Simplifying gives

`t^2+2+(1)/(t^2)=9,`

`t^2+(1)/(t^2)=7.`

The question asks for
`t^5+(1)/(t^5),` so we can square the equation again and simplify to get higher powers into the expression:

`(t^2+(1)/(t^2))^2=49,`

`t^4+(2 \cdot t^2 \cdot (1)/(t^2)) + (1)/(t^4)=49,`

`t^4+2+ (1)/(t^4)=49,`

`t^4+(1)/(t^4)=47.`

Multiplying this expression by
`t+(1)/(t)` to try and get a fifth power gives

`(t^4+(1)/(t^4))(t+(1)/(t))=t^5+t^3+(1)/(t^3)+(1)/(t^5).`

The only thing left we need is
`t^3+(1)/(t^3)` to subtract from this; we know everything else. Since
`t^3+(1)/(t^3)` can be written as
`(t+(1)/(t))(t^2-1+(1)/(t^2)),` we can simply plug in the values we know for
`t+(1)/(t)` and
`t^2+(1)/(t^2):`

`t^3+(1)/(t^3)=(3)(7-1)=18.`

All that is left is to plug it in our equation here:

`(t^4+(1)/(t^4))(t+(1)/(t))=t^5+t^3+(1)/(t^3)+(1)/(t^5),`

`(47)(3)=t^5+(1)/(t^5)+18,`

Multiplying and rearranging gives:

`t^5+(1)/(t^5)=123.`