Question

please answer correctly ​

Answer 1

Step-by-step explanation:

We can factorize a difference of 7th powers using the identity (easily verified by expanding the brackets):

a7−b7=(a−b)(a6+a5b+a4b2+a3b3+a2b4+ab5+b6)

Similarly, taking x=a and 1/x=b, we can write the desired expression:

x7−1x7=(x−1x)(x6+x4+x2+1+1x2+1x4+1x6) ……(1)

We then need to calculate the values of three different partial sums:

x2+1x2, x4+1x4, and x6+1x6

Squaring the given equation, x-(1/x)=3, we have:

(x−1x)2=9=x2−2+1x2⟹9

x2+1x2=11. eq2

⟹x4+1x4=119. eq3

Multiplying above expressions (2) and (3) yields the sum of 6th powers:

(x2+1x2)(x4+1x4)=11*119

⟹x6+1x6+x2+1x2=11*119

⟹x6+1x6=1309. eq4

Replacing (2), (3), and (4) into (1), we have:

x7−1x7=4(1309+119+11+1)=4*1440= 5760