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King's Maths School

Challenge 385: Algebra-Assisted Arithmetic

How can this algebra help you to avoid some very long multiplication or division?

(i) Show that

1+x2+(1+x)2=2(1+x)2−2x1 + x^2 + (1 + x)^2 = 2(1+x)^2 - 2x

(ii) show that

(1+x2+(1+x)2)2=2(1+x4+(1+x)4)(1 + x^2 + (1 +x)^2)^2 = 2(1 + x^4 + (1 + x)^4)

Hence evaluate, without using any technology, and showing working if necessary,

1+16004+160141+16002+16012\frac{1 + 1600^4 + 1601^4}{1 + 1600^2 + 1601^2}