Question

if x+1/x=4 then findx^5/1/x^5​

Answer 1

Answer :

x^5 + 1/x^5 = 724

Solution :

• Given : x + 1/x = 4
• To find : x^5 + 1/x^5 = ?

We have ,

x + 1/x = 4 ------(1)

Now ,

Squaring both the sides of eq-(1) , we get ;

=> (x + 1/x)² = 4²

=> x² + (1/x²) + 2•x•(1/x) = 16

=> x² + 1/x² + 2 = 16

=> x² + 1/x² = 16 - 2

=> x² + 1/x² = 14 ------(2)

Now ,

Cubing both the sides of eq-(1) , we get ;

=> (x + 1/x)³ = 4³

=> x³ + (1/x)³ + 3•x•(1/x)•(x + 1/x) = 64

=> x³ + 1/x³ + 3•1•4 = 64

=> x³ + 1/x³ + 12 = 64

=> x³ + 1/x³ = 64 - 12

=> x³ + 1/x³ = 52 -------(3)

Now ,

Multiplying eq-(2) and (3) , we get ;

=> (x² + 1/x²)(x³ + 1/x³) = 14•52

=> x²•x³ + x²/x³ + x³/x² + 1/(x²•x³) = 728

=> x^5 + 1/x + x + 1/x^5 = 728

=> (x^5 + 1/x^5) + (x + 1/x) = 728

=> x^5 + 1/x^5 + 4 = 728

=> x^5 + 1/x^5 = 728 - 4

=> x^5 + 1/x^5 = 724

Hence ,

x^5 + 1/x^5 = 724