Your solution is more rigorous but here's a quick alternative. Also I would have wanted to use a calculator to evaluate the arithmetic that led to xy = 17 but it wasn't allowed. It's unlikely that the √2 will not be in the solution. If there is a solution, it's probably going to be of the form a + b√2. So (a + b√2)^3 = 55 + 63√2 Expanding a^3 + 3a^2*b√2 + 3ab^2*2 + b^3*2√2 = 55 + 63√2 For integer a & b, this means that a^3 + 6ab^2 = 55 Factoring a(a^2 + 6b^2) = 55 The terms inside the parentheses will be positive, so a must be >0. Also a must be < 55^(1/3), which means that a = 1, 2, or 3. The a, (a^2 + 6b^2) possibilities are 1,55 or 5,11 etc. The only one that works is a=1. Then b=3. The other half of the first expansion is 2b^3√2 + 3a^2*b√2 = 63√2 Eliminating the √2 gives 2b^3 + 3a^2*b = 63 Or b*(2b^2 + 3a^2) = 63 Like before b must be >0 and
Посчитать кубический корень из 4913 без калькулятора. Нет ничего проще, конечно же - 17. Удивительно, почему он сразу ответ без калькулятора не написал.
Your solution is more rigorous but here's a quick alternative. Also I would have wanted to use a calculator to evaluate the arithmetic that led to xy = 17 but it wasn't allowed.
It's unlikely that the √2 will not be in the solution. If there is a solution, it's probably going to be of the form a + b√2.
So (a + b√2)^3 = 55 + 63√2
Expanding a^3 + 3a^2*b√2 + 3ab^2*2 + b^3*2√2 = 55 + 63√2
For integer a & b, this means that a^3 + 6ab^2 = 55
Factoring a(a^2 + 6b^2) = 55
The terms inside the parentheses will be positive, so a must be >0. Also a must be < 55^(1/3), which means that a = 1, 2, or 3.
The a, (a^2 + 6b^2) possibilities are 1,55 or 5,11 etc. The only one that works is a=1. Then b=3.
The other half of the first expansion is 2b^3√2 + 3a^2*b√2 = 63√2
Eliminating the √2 gives 2b^3 + 3a^2*b = 63
Or b*(2b^2 + 3a^2) = 63 Like before b must be >0 and
Посчитать кубический корень из 4913 без калькулятора. Нет ничего проще, конечно же - 17. Удивительно, почему он сразу ответ без калькулятора не написал.