1. If `x+1/x = 2`, then find `x^2+1/x^2`

Solution:
Here `x+1/x=2`


Now, We know that
`x^2+1/x^2=(x+1/x)^2-2`

`:.x^2+1/x^2=2^2-2`

`:.x^2+1/x^2=4-2`

`:.x^2+1/x^2=2`

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2. If `x+1/x = 2`, then find `x^3+1/x^3`

Solution:
Here `x+1/x=2`


Now, We know that
`x^3+1/x^3=(x+1/x)^3-3(x+1/x)`

`:.x^3+1/x^3=2^3-3*2`

`:.x^3+1/x^3=8-6`

`:.x^3+1/x^3=2`

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3. If `x+1/x = 2`, then find `x^4+1/x^4`

Solution:
Here `x+1/x=2`


Now, We know that
`x^2+1/x^2=(x+1/x)^2-2`

`:.x^2+1/x^2=2^2-2`

`:.x^2+1/x^2=4-2`

`:.x^2+1/x^2=2`


Now, We know that
`(x^4+1/x^4)=(x^2+1/x^2)^2-2`

`:.(x^4+1/x^4)=2^2-2`

`:.(x^4+1/x^4)=4-2`

`:.x^4+1/x^4=2`

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4. If `x+1/x = 2`, then find `x-1/x`

Solution:
Here `x+1/x=2`


Now, We know that
`(x-1/x)^2=(x+1/x)^2-4`

`:.(x-1/x)^2=2^2-4`

`:.(x-1/x)^2=4-4`

`:.(x-1/x)^2=0`

`:.x-1/x=0`

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5. If `x+1/x = 2`, then find `x^2-1/x^2`

Solution:
Here `x+1/x=2`


Now, We know that
`(x-1/x)^2=(x+1/x)^2-4`

`:.(x-1/x)^2=2^2-4`

`:.(x-1/x)^2=4-4`

`:.(x-1/x)^2=0`

`:.x-1/x=0`


Now, We know that
`(x^2-1/x^2)=(x-1/x) (x+1/x)`

`:.(x^2-1/x^2)=0*2`

`:.x^2-1/x^2=0`

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6. If `x+1/x = 2`, then find `x^3-1/x^3`

Solution:
Here `x+1/x=2`


Now, We know that
`(x-1/x)^2=(x+1/x)^2-4`

`:.(x-1/x)^2=2^2-4`

`:.(x-1/x)^2=4-4`

`:.(x-1/x)^2=0`

`:.x-1/x=0`


Now, We know that
`x^3-1/x^3=(x-1/x)^3+3(x-1/x)`

`:.x^3-1/x^3=0^3+3*0`

`:.x^3-1/x^3=0+0`

`:.x^3-1/x^3=0`

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7. If `x+1/x = 2`, then find `x^4-1/x^4`

Solution:
Here `x+1/x=2`


Now, We know that
`(x-1/x)^2=(x+1/x)^2-4`

`:.(x-1/x)^2=2^2-4`

`:.(x-1/x)^2=4-4`

`:.(x-1/x)^2=0`

`:.x-1/x=0`


Now, We know that
`(x^2-1/x^2)=(x-1/x) (x+1/x)`

`:.(x^2-1/x^2)=0*2`

`:.x^2-1/x^2=0`


Now, We know that
`x^2+1/x^2=(x+1/x)^2-2`

`:.x^2+1/x^2=2^2-2`

`:.x^2+1/x^2=4-2`

`:.x^2+1/x^2=2`


Now, We know that
`(x^4-1/x^4)=(x^2-1/x^2) (x^2+1/x^2)`

`:.(x^4-1/x^4)=0*2`

`:.x^4-1/x^4=0`

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This material is intended as a summary. Use your textbook for detail explanation.
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