TO start with the topic first of all its necessary to know basic identities:-

a + b whole square is what? I.e a square +2ab+ b square. Similarly there are many more. TO begin with our basic topic we will try some questions and their approaches.

Q1. IF x +1/x = 4 then what is x² +1/x²

So guys we are given value of x + 1/x =4. Now simply square both sides. You will get x² + 2 + 1/x² as 16. Now if you take to on RHS side it will be simply 16-2 i.e 14. Hence 14 is your answer. So dear reader you can follow a shortcut to solve orally that 4 square minus 2. I.e same 16-2 . Similarly if you have 9 instead of 4 in the question the answer would be 81-2 =79.

If now instead of this you need to find value of x⁴ +1/x⁴ . Then how would you solve. Just take out value of earlier one and square it and minus. For eg in first question as 14 was our answer subtract 2 from it and its your new answer .

Q2. If now in first question you need to find x3 +1/x3 (note its cube not three)

So you can use a shortcut i.e. p3 -3p. I.e. 4 cube -4×3 which will be 64-12=52.

Q3. If x-1/x =p then x3-1/x3 will be

Its easy just in above formula replace minus sign by plus . p3+3p.

Q4. If in first question I need to find x5+1/x5 then.

Here you need to know just the basic formula (x^a )^b = x^a+b . Dear reader we now that 2+3=5 . So find the value in example 1 and 2. I.e 14 and 52 now multiply both of them and minus with value in question. 728-4=724.

For reference

https://www.google.com/url?sa=t&source=web&rct=j&url=https://byjus.com/maths/algebraic-identities/&ved=2ahUKEwiXnezZueLxAhWIxzgGHfbGChQQFnoECDMQAQ&usg=AOvVaw3kOAhEDrCdK4_MLjcyXg9U

https://www.google.com/url?sa=t&source=web&rct=j&url=https://www.vedantu.com/maths/algebraic-identities&ved=2ahUKEwiXnezZueLxAhWIxzgGHfbGChQQFnoECDQQAQ&usg=AOvVaw3S4Vj2T2sw7UHmGmefLXFC