perplexus.info :: Just Math : Square-sum fraction

Square-sum fraction (Posted on 2024-06-04)

If x²/y² + y²/x² = 254, then find the value of x⁵/y⁵ + y⁵/x⁵.

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solution

| Comment 1 of 4

(x^2/y^2 + y^2/x^2) = 254 ***

x/y + y/x = k

x^2/y^2 + 2 + y^2/x^2 = k^2

(x/y + y/x)^2 = 254+2 = 256

(x/y + y/x) = 16 ***

(x/y + y/x) * (x^2/y^2 + y^2/x^2) = 16 * 254 = 4,064

= (x^3/y^3 + y^3/x^3) + (x/y + y/x)

(x^3/y^3 + y^3/x^3) = 4,064 - (x/y + y/x) = 4,048

(x^3/y^3 + y^3/x^3) = 4,048 ***

x^2/y^2 + y^2/x^2 = 254

x^4/y^4 + 2 + y^4/x^4 = 254^2

(x^4/y^4 + y^4/x^4) = 254^2 - 2 = 64,514

(x^4/y^4 + y^4/x^4) = 64,514 ***

(x/y + y/x) * (x^4/y^4 + y^4/x^4) = 16 * 64,514

= (x^5/y^5 + y^5/x^5) + (x^3/y^3 + y^3/x^3)

(x^5/y^5 + y^5/x^5) = 16 * 64,514 - (x^3/y^3 + y^3/x^3)

(x^5/y^5 + y^5/x^5) = 1,028,176

Posted by Larry on 2024-06-04 20:22:13

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