Question

If x + 1/x = 3, then,
x^7 + 1/x^7 = ?

Answer 1

Answer:

Assuming the question is “If x-(1/x)=4, then what is x^7-(1/x)^7=?” as otherwise it would be a very easy problem.

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

$a7−b7=(a−b)(a6+a5b+a4b2+a3b3+a2b4+ab5+b6)</p><p>$

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

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

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

$x2+1x2, x4+1x4, and x6+1x6</p><p>$

Squaring the given equation, x-(1/x)=4, we have:

$(x−1x)2=16=x2−2+1x2⟹(x−1x)2=16=x2−2+1x2⟹</p><p></p><p>x2+1x2=18x2+1x2=18…………………..……….(2)</p><p></p><p>⟹x4+1x4=322⟹x4+1x4=322……………….(3)</p><p></p><p>$

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

$(x2+1x2)(x4+1x4)=18×322(x2+1x2)(x4+1x4)=18×322</p><p></p><p>⟹</p><p></p><p>$

$⟹x6+1x6+x2+1x2=18×322$

⟹x6+1x6=5778

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

$x7−1x7=4(5778+322+18+1)=2447$