Известно, что число а = 1+корень5 / 2 называют з...

Translating and interpreting math expression

I'm trying to decode the math problem in Russian. It mentions the golden ratio, φ = (1+√5)/2, which is a root of the equation x² - x - 1 = 0. The question asks for the numeric value of the expression a^10 - 55a - 14. There’s a case of possible typo, and I’ll use known Fibonacci identities to simplify the powers of φ.

Simplifying the expression

Okay, so using the Fibonacci identity for φ^10 = 55φ + 34, when I substitute into the expression a^{10} - 55a - 14, it simplifies nicely to 20. I double-checked with the exact numeric computations and confirmed the result is accurate: φ^10 = 122.991869... and the numeric value comes out cleanly as 20. Looks good!

Providing a brief explanation

I’ll calculate the result step-by-step while avoiding too much reasoning in detail. The identity φ^n = F_n φ + F_{n-1} helps simplify the expression, and the final numeric result comes out to 20. I'll proceed with this answer in Russian, correcting the typo in the equation to x² - x - 1 = 0.

Thought for 18s