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Welcome to NRICH.

x⁴ + 1/x⁴ from x + 1/x

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By Anonymous on Friday, November 3, 2000 - 11:24 am:

Hi, I've come across this problem, which seems easy (and is probably easy) but I cannot solve it:

If x +  1 x = 8, then what is x4 +  1 x4 equal to and why?

Thank you very much

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By William Astle (Wja24) on Friday, November 3, 2000 - 11:35 am:

Consider

æ è x +  1 x ö ø 4

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By William Astle (Wja24) on Friday, November 3, 2000 - 11:38 am:

and also perhaps you might like to look at

æ è x +  1 x ö ø 2

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By Sasha Muzyrya (P3164) on Tuesday, November 7, 2000 - 04:17 pm:

hello, I don't know how to solve that algebra problem either

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By Michael Doré (Md285) on Tuesday, November 7, 2000 - 04:26 pm:

OK, well let's follow William's guidelines (in reverse order).

x + 1/x = 8

So we know that:

(x + 1/x)² = 64

Now expand this out:

x² + 2 + 1/x² = 64

x² + 1/x² = 62

We'll need that later. Anyway now look at (x + 1/x)⁴:

(x + 1/x)⁴ = 64²
x⁴ + 4x² + 6 + 4/x² + 1/x⁴ = 64²

So:

x⁴ + 1/x⁴ + 4(x² + 1/x²) = 64² - 6

x⁴ + 1/x⁴ + 4×62 = 64² - 6

And finally:

x⁴ + 1/x⁴ = 64² - 4×62 - 6
x⁴ + 1/x⁴ = 3842

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