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Ҳисоб кунед:

\(\sqrt[3]{50+19\sqrt{7}}\)

Бигзор \(50+19\sqrt{7}=(a+b\sqrt{7})^3\)

\(50+19\sqrt{7}=a^3+3\sqrt{7}a^2b+21ab^2+7\sqrt{7}b^3\)

\(50+19\sqrt{7}=(a^3+21ab^2)+\sqrt{7}\cdot(3a^2b+7b^3)\)

\(\left\{ \begin{array}{rcl} a^3+21ab^2=50\\3a^2b+7b^3=19 \end{array}\right.\)

\(\frac{a^3+21ab^2}{3a^2b+7b^3}=\frac{50}{19}\)

\(\frac{a^3+21ab^2}{3a^2b+7b^3}=\frac{\frac{a^3}{b^3}+21\frac{a}{b}}{3\frac{a^2}{b^2}+7}\)

\(t=\frac{a}{b}\)

\(\frac{t^3+21t}{3t^2+7}=\frac{50}{19}\)

\(19t^3-150t^2+399t-350=0\)

Муодилаи кубиро ҳал карда решҳояшро меёбем.

Решаи ин муодилаи кубӣ t = 2 мебошад.

\(t=\frac{a}{b},\)

\(t=2\)

\(a=2b\)

\(a^3+21ab^2=8b^3+42b^3=50\)

\(50b^3=50\)

\(b^3=1\)

\(b=1\)

\(a=2\)

\(50+19\sqrt{7}=(a+b\sqrt{7})^3\)

\(50+19\sqrt{7}=(2+\sqrt{7})^3\)

\(\sqrt[3]{50+19\sqrt{7}}=2+\sqrt{7}\)

Ҷавоб: \(2+\sqrt{7}\)