التمارين المفصلة

تبسيط A:
\( A = -\sqrt{5} + \left[ a - \frac{13}{2} - (-1.5 - b) \right] \)
\( = -\sqrt{5} + a - \frac{13}{2} + 1.5 + b \)
\( = -\sqrt{5} + a - \frac{13}{2} + \frac{3}{2} + b \)
\( = a + b - \sqrt{5} - \frac{10}{2} \)
\( = a + b - \sqrt{5} - 5 \)

تبسيط B:
\( B = (-3 + a) - \left[ \sqrt{5} + \frac{4}{5} - (-5 - b) \right] \)
\( = (-3 + a) - \left[ \sqrt{5} + \frac{4}{5} + 5 + b \right] \)
\( = -3 + a - \sqrt{5} - \frac{4}{5} - 5 - b \)
\( = a - b - \sqrt{5} - 8 - \frac{4}{5} \)
\( = a - b - \sqrt{5} - \frac{40}{5} - \frac{4}{5} \)
\( = a - b - \sqrt{5} - \frac{44}{5} \)

\( A = a + b - \sqrt{5} - 5 \)
\( B = a - b - \sqrt{5} - \frac{44}{5} \)

\( B - A = \left(a - b - \sqrt{5} - \frac{44}{5}\right) - \left(a + b - \sqrt{5} - 5\right) \)
\( = a - b - \sqrt{5} - \frac{44}{5} - a - b + \sqrt{5} + 5 \)
\( = -2b - \frac{44}{5} + 5 \)
\( = -2b - \frac{44}{5} + \frac{25}{5} \)
\( = -2b - \frac{19}{5} \)

\( B = a - b - \sqrt{5} - \frac{44}{5} \)

بالتعويض: \( a = \sqrt{5} - 2 \) و \( b = \sqrt{5} + 1 \)

\( B = (\sqrt{5} - 2) - (\sqrt{5} + 1) - \sqrt{5} - \frac{44}{5} \)
\( = \sqrt{5} - 2 - \sqrt{5} - 1 - \sqrt{5} - \frac{44}{5} \)
\( = -3 - \sqrt{5} - \frac{44}{5} \)
\( = -\frac{15}{5} - \sqrt{5} - \frac{44}{5} \)
\( = -\frac{59}{5} - \sqrt{5} \)

\( B = a - b - \sqrt{5} - \frac{44}{5} = -3 \)

\( a - b - \sqrt{5} - \frac{44}{5} = -3 \)
\( a - b = -3 + \sqrt{5} + \frac{44}{5} \)
\( a - b = -\frac{15}{5} + \sqrt{5} + \frac{44}{5} \)
\( a - b = \frac{29}{5} + \sqrt{5} \)

العلاقة: \( a - b = \frac{29}{5} + \sqrt{5} \)

\( A = -\sqrt{2} - (x - \sqrt{2}) - [4 - (y + \sqrt{2})] \)
\( = -\sqrt{2} - x + \sqrt{2} - 4 + y + \sqrt{2} \)
\( = -x + y - 4 + (-\sqrt{2} + \sqrt{2} + \sqrt{2}) \)
\( = -x + y - 4 + \sqrt{2} \)

\( A = -x + y - 4 + \sqrt{2} = -11 \)
\( -x + y = -11 + 4 - \sqrt{2} \)
\( -x + y = -7 - \sqrt{2} \)
\( x - y = 7 + \sqrt{2} \)

\( A = -x + y - 4 + \sqrt{2} \)
\( = -3 + (-\sqrt{2}) - 4 + \sqrt{2} \)
\( = -3 - \sqrt{2} - 4 + \sqrt{2} \)
\( = -7 \)

تبسيط E:
\( E = -(1 - a) + \sqrt{3} - [-b - (1 - a)] - (-2 - a) \)
\( = -1 + a + \sqrt{3} - (-b - 1 + a) + 2 + a \)
\( = -1 + a + \sqrt{3} + b + 1 - a + 2 + a \)
\( = a + b + \sqrt{3} + 2 \)

تبسيط F:
\( F = -b - [\sqrt{3} - (1 - a) - b] - (b - 2) \)
\( = -b - (\sqrt{3} - 1 + a - b) - b + 2 \)
\( = -b - \sqrt{3} + 1 - a + b - b + 2 \)
\( = -a - b - \sqrt{3} + 3 \)

بما أن a = 2 - b:

حساب E:
\( E = a + b + \sqrt{3} + 2 \)
\( = (2 - b) + b + \sqrt{3} + 2 \)
\( = 2 + \sqrt{3} + 2 \)
\( = 4 + \sqrt{3} \)

حساب F:
\( F = -a - b - \sqrt{3} + 3 \)
\( = -(2 - b) - b - \sqrt{3} + 3 \)
\( = -2 + b - b - \sqrt{3} + 3 \)
\( = 1 - \sqrt{3} \)

إذا كان a و b متقابلان فإن: \( a = -b \)

\( E = a + b + \sqrt{3} + 2 \)
\( = (-b) + b + \sqrt{3} + 2 \)
\( = \sqrt{3} + 2 \)

إذا كان F و \( \pi - \sqrt{3} \) متقابلان فإن:
\( F + (\pi - \sqrt{3}) = 0 \)
\( F = -(\pi - \sqrt{3}) = -\pi + \sqrt{3} \)

\( -a - b - \sqrt{3} + 3 = -\pi + \sqrt{3} \)
\( -a - b + 3 = -\pi + \sqrt{3} + \sqrt{3} \)
\( -a - b + 3 = -\pi + 2\sqrt{3} \)
\( a + b = 3 + \pi - 2\sqrt{3} \)

تبسيط E:
\( E = -\pi - \left[\left(-\frac{2}{5}\right) + (-b - 1.5)\right] + (1.2 + a) \)
\( = -\pi - \left(-\frac{2}{5} - b - 1.5\right) + 1.2 + a \)
\( = -\pi + \frac{2}{5} + b + 1.5 + 1.2 + a \)
\( = a + b - \pi + \frac{2}{5} + 2.7 \)
\( = a + b - \pi + 0.4 + 2.7 \)
\( = a + b - \pi + 3.1 \)

تبسيط F:
\( F = 1 - \left[a + \frac{3}{2}\right] - (\pi + b) \)
\( = 1 - a - \frac{3}{2} - \pi - b \)
\( = -a - b - \pi + 1 - 1.5 \)
\( = -a - b - \pi - 0.5 \)

إذا كان E مقابل (2-b) فإن: \( E = -(2-b) = -2 + b \)

\( E = a + b - \pi + 3.1 = -2 + b \)
\( a + b - \pi + 3.1 = -2 + b \)
\( a - \pi + 3.1 = -2 \)
\( a = -2 + \pi - 3.1 \)
\( a = \pi - 5.1 \)

\( F = -a - b - \pi - 0.5 \)

بالتعويض: \( b = \sqrt{3} - a \)

\( F = -a - (\sqrt{3} - a) - \pi - 0.5 \)
\( = -a - \sqrt{3} + a - \pi - 0.5 \)
\( = -\sqrt{3} - \pi - 0.5 \)
\( = -\sqrt{3} - \pi - \frac{1}{2} \)