Prove the following identity involving the golden ratio (see picture)?

Question

Duke · Accepted Answer

We need to prove the following equivalent identity:
√ (∛(34Φ + 21) - ∛(34Φ - 55) ) = ∛(5√2 + 7) - ∛(5√2 - 7)
Indeed both sides now are equal to 2. Left hand side:
34Φ + 21 = 17√5 + 38 = 5√5 + 30 + 12√5 + 8 = (√5 + 2)³;
34Φ - 55 = 17√5 - 38 = 5√5 - 30 + 12√5 - 8 = (√5 - 2)³;
√ (∛(34Φ + 21) - ∛(34Φ - 55) ) = √( (√5 + 2) - (√5 - 2) ) = √4 = 2
Right hand side:
5√2 + 7 = 2√2 + 6 + 3√2 + 1 = (√2 + 1)³;
5√2 - 7 = 2√2 - 6 + 3√2 - 1 = (√2 - 1)³;
∛(5√2 + 7) - ∛(5√2 - 7) = (√2 + 1) - (√2 - 1) = 2 as required.

Prove the following identity involving the golden ratio (see picture)?

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