Question

Calcule, passo a passo, a expressão:
{[(9+v17)/4]-[(9-v17)/4)]}^2

Obs.: Mesma expressão na foto; v17 = raiz quadrada de 17; ^2 = elevado ao quadrado.

Answer 1

$\left(\dfrac{9+\sqrt{17}}{4}-\dfrac{9-\sqrt{17}}{4} \right )^{2}\\ \\ \\ =\left(\dfrac{9+\sqrt{17}-(9-\sqrt{17})}{4} \right )^{2}\\ \\ \\ =\left(\dfrac{9+\sqrt{17}-9+\sqrt{17}}{4} \right )^{2}\\ \\ \\ =\left(\dfrac{\diagup\!\!\!\! 9-\diagup\!\!\!\! 9+\sqrt{17}+\sqrt{17}}{4} \right )^{2}\\ \\ \\ =\left(\dfrac{2\sqrt{17}}{4} \right )^{2}\\ \\ \\ =\left(\dfrac{\diagup\!\!\!\! 2\sqrt{17}}{\diagup\!\!\!\! 2\cdot 2} \right )^{2}\\ \\ \\ =\left(\dfrac{\sqrt{17}}{2} \right )^{2}\\ \\ \\ =\dfrac{(\sqrt{17})\,^{2}}{2^{2}}\\ \\ \\ =\dfrac{17}{4}\\ \\ \\ \\ \boxed{ \begin{array}{c} \left(\dfrac{9+\sqrt{17}}{4}-\dfrac{9-\sqrt{17}}{4} \right )^{2}=\dfrac{17}{4} \end{array} }$