Expressions & Identities (Squares & Powers). What is Integer ?

---

Formula: How to Identify Category

Reciprocal Power Identities & Half-Power Shortcut

Base Formula - A

If, a + 1/a = x

• Square: a²+1/a² = x²- 2


• Fourth Power: a⁴+1/a⁴ = (x²- 2)²- 2

Base Formula - B

If, aⁿ + 1/aⁿ = k

Then

a^n/2 - 1/a^n/2 = √k-2

Instant Summary

1. If given sign is “+” and required sign is “+”

aⁿ + 1/aⁿ ⇒ a^n/2 + 1/a ^n/2

Formula: √(k + 2)
Memory Trick: Plus → Add 2

2.  If given sign is “+” and required sign is “−“

aⁿ + 1/aⁿ ⇒ a^n/2 - 1/a^n/2

Formula: √(k − 2)
Memory Trick: Minus → Subtract 2

3.  If given sign is “–“ and required sign is “+”

aⁿ - 1/aⁿ ⇒ a^n/2 + 1/a^n/2

Formula: √(k + 2)
Memory Trick: Plus → Add 2

4.  If given sign is “–“ and required sign is “−“

aⁿ - 1/aⁿ ⇒ a^n/2 - 1/a ^n/2

   Formula: √(k - 2)
   Memory Trick: Minus → Subtract 2

Super Shortcut Memory : Half power = √( K ± 2 )

• If required sign is "+" → add 2


• If required sign is " −"→ subtract 2

Q. If a+1/a=4, then find the value of a⁴+1/a⁴

Options: 204, 196, 198, 194 

Trick: Given: a + 1/a = x

Then: 1^st: a² + 1/a² = x²-2,  2^nd: a⁴ + 1/a⁴ = (x²−2)²−2

1^st Square it: (4)² = 16, Then subtract 2: a²+1/a² = 16−2 =14

2^nd Again square 14: 14² = 196, Then subtract 2: a⁴+1/a⁴ = 196-2 = 194

Ans: 194

Note : First square → minus 2, Again square → minus 2

Tricks : Check Base Formula - B:- 1, 2, 3 & 4 nos on Top

Q. If x²⁵+1/x²⁵ = 627 then find: x^12.5-1/x^12.5  = ?

Options: (a) 24 (b) 25 (c) 125 (d) 26

Solⁿ

X^12.5- x^1/12.5​ = √627- 2​ =√625 = 25

Ans: 25

Q. If x¹⁰+1/x¹⁰ = 194 then find: x⁵+1/x⁵  = ?

Options: (a) 12 (b) 13 (c) 14 (d) 16

Solⁿ

X⁵- x^1/5​ = √194+2​ =√196 = 14

Ans: 14

Q. Solve (1/7)^-4 + (1/9)^-4  + (1/5)^-4 = ?

Solⁿ

Trick : (1/a)^-n = aⁿ

= (1/7)^-4 + (1/9)^-4  + (1/5)^-4 

= 7⁴+9⁴+5⁴

= 2401+6561+625

= 15760

 Ans: 15760

Q. What is Integer ?

Definition: An integer is a whole number which can be positive, negative, or zero, without any fractional or decimal part. Integer বুলিলে সেই পূৰ্ণসংখ্যাক বুজায় যি ধনাত্মক, ঋণাত্মক বা শূন্য হ’ব পাৰে, আৰু ইয়াত কোনো ভগ্নাংশ বা দশমিক অংশ নাথাকে। Ex / উদাহৰণ:: …, -3, -2, -1, 0, 1, 2, 3, …

Note: 0 is an integer. Fractions (1/2, 3/4) are not integers . Decimals (2.5, -1.2) are not integers

Q.  If a - 1/a = 7 then what the value of a⁵ - 1/a⁵ = ?

Options: (a) 18557 (b) 17147 (c) 14557 (d) 21107

Trick: Whenever a-1/a = x, (x is integer), the value of aⁿ-1/aⁿ must be divisible by x.

• Option (a) 18557 - divide by 7


• 18557 ÷ 7 = 2651 integer - correct


• If it is not divisible by 7, the answer is wrong.

Ans : Option (a) 18557