Solving Questions of Fractions and Exponential Equations

 If X + 1/X = 10 then X⁵ + 1/X⁵ = ?

Solution :- 

Trick :- Coefficient is summation of Above two numbers

For example after squaring we get equation like this à X² + 2 +1/X²

Coefficients are 1  2  1

For cube coefficients can be derived as 1 (1+2=3) (2+1=3) 1

For 4^th power coefficients can be derived as 1 (1+3=4) (3+3=6) (3+1=4) 1

1
1	2	1
1	3	3	1
1	4	6	4	1
1	5	10	10	5	1
1	6	15	20	15	6	1
1	7	21	35	35	21	7	1
1	8	28	56	70	56	28	8	1
1	9	36	84	126	126	84	36	9	1

 

X + 1/X = 10 ………… (1)

Squaring on both sides 

(X + 1/X)² = 100    so X² + 2 +1/X² = 100 hence X² + 1/X² = 98 ………(2)

Similarly

(X + 1/X)³ = 100    so X³+ 3*X² * 1/X + 3*X * 1/X² +1/X³ = 1000

Therefore, X³+ 3X + 3/X +1/X³ = 1000 

X³+1/X³ +3(X+1/X) = 1000

X³+1/X³ +3(10) = 1000

X³+1/X³ = 970 ……………(3)

Now if You closely observed even powers terms are missing as they are missing which means we don’t need to worry about calculating value of X⁴+1/X⁴ for X⁵+1/X⁵ so we directly jump on 5^th power equation.

(X + 1/X)⁵ = 10⁵   

X⁵+ 5*X⁴ * 1/X + 10*X³ * 1/X² + 10*X² * 1/X³ + 5*X * 1/X⁴ +1/X⁵ = 100000

X⁵+ 5X³ + 10X + 10/X + 5/X³ +1/X⁵ = 100000

X⁵+ 1/X⁵ + 5X³ +  5/X³ + 10X + 10/X = 100000

X⁵+ 1/X⁵ + 5(X³ + 1/X³) + 10(X + 1/X) = 100000

X⁵+ 1/X⁵ + 5(970) + 10(10) = 100000  ……… from equation 1 and 3

X⁵+ 1/X⁵ = 95050.