Question

If , then what is the exact value of x5+1/x5 for x≥0?

Answer 1

 

Expand the square

(x+1x)2=x2+1x2+2x1x Simplify the term 2x1x to 2

=x2+1x2+2

Use x2+1x2=10 to simplify the above and obtain

(x+1x)2=10+2=12

Take the square root of both sides

(x+1x)=√12=2√3

One way to obtain terms with x3 and 1x3, expand the following

(x+1x)(x2+1x2)=x3+1x+x+1x3

Deduce x3+1x3 from the above in terms of known quantities

x3+1x3=(x+1x)(x2+1x2)−(x+1x)

Substitute the known quantities by their numerical values

x3+1x3=2√3×10−2√3=18√3

One way to obtain terms with x5 and 1x5, expand the following

 

(x3+1x3)(x2+1x2)=x5+x+1x+1x5

Deduce x5+1x5 from the above in terms of known quantities

x5+1x5=(x3+1x3)(x2+1x2)−(x+1x)

Substitute known quantities by their numerical values

x5+1x5=18√3×10−2√3=178√3

 

178√3