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Let w be the real number w = cbrt(20 + 14√2) + cbrt(20 - 14√2), where…
Question
Let w be the real number w = cbrt(20 + 14√2) + cbrt(20 - 14√2), where cbrt denotes the real cube root. Find w.
✓ Verified answer: 4checked by our engine — not a guess
Step-by-step solution
Let p = cbrt(20+14√2) and q = cbrt(20-14√2), so w = p + q. Then
p^3 + q^3 = (20+14√2)+(20-14√2) = 40, and pq = cbrt((20)^2 - (14√2)^2) = cbrt(400 - 392) = cbrt(8) = 2.
Using w^3 = p^3 + q^3 + 3pq(p+q) = 40 + 3·2·w = 40 + 6w, we get w^3 - 6w - 40 = 0.
Testing w = 4: 64 - 24 - 40 = 0. Since the function increases for w>√2 and w is clearly positive, w = 4 is the unique real value.
Final answer4
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