If x > 0, and x4 + = 2207, what is the value of x7 + ?
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Correct answer: B
Explanation
The correct answer is option 2 i.e. 710647.
Formula used:
(a + b)2 = a2 + b2 + 2ab ---- (1)
(a + b)3 = a3 + b3 + 3ab(a + b) ---- (1)
Calculations:
According to the question;
⇒ x4 + 1/x4 + 2 = (x2 + 1/x2)2
⇒ 2207 + 2 = (x2 + 1/x2)2
⇒ √2209 = (x2 + 1/x2)
⇒ (x2 + 1/x2) = 47
Similarly, we have
⇒ (x + 1/x)2 = x2 + 1/x2 + 2
⇒ (x + 1/x)2 = 47 + 2 = 49
⇒ (x + 1/x) = 7
Now using equation (2), we get
⇒ (x + 1/x)3 = x3 + 1/x3 + 3(x + 1/x)
⇒ 343 = x3 + 1/x3 + 21
⇒ x3 + 1/x3 = 322
Hence, x7 + 1/x7 = (x3 + 1/x3)(x4 + 1/x4) - (x + 1/x)
⇒ x7 + 1/x7 = (322)(2207) - 7 = 710647