§2. Приложения математики в реальной жизни — Дополнительные упражнения к параграфу 1 — 90 — стр. 30

(а)
\[
(\sqrt{2}+\sqrt{3})\cdot(\sqrt{2}-\sqrt{3})=(\sqrt{2})^2-(\sqrt{3})^2=2-3=-1 \in \mathbb{Q}
\]
Рациональное число
(б)
\[
(\sqrt{2}+2\sqrt{3})\cdot(\sqrt{2}-\sqrt{3})=(\sqrt{2})^2+2\sqrt{6}-\sqrt{6}-2(\sqrt{3})^2=2+\sqrt{6}-6=\sqrt{6}-4 \notin \mathbb{Q}
\]
Иррациональное число
(в)
\[
\frac{1}{2+\sqrt{3}}+\frac{1}{2-\sqrt{3}}=\frac{2-\sqrt{3}+2+\sqrt{3}}{(2+\sqrt{3})(2-\sqrt{3})}=\frac{4}{4-3}=4 \in \mathbb{Q}
\]
Рациональное число
(г)
\[
\frac{1}{\sqrt{3}-\sqrt{2}}-\frac{1}{\sqrt{3}+\sqrt{2}}=\frac{\sqrt{3}+\sqrt{2}-(\sqrt{3}-\sqrt{2})}{(\sqrt{3}-\sqrt{2})(\sqrt{3}+\sqrt{2})}=\frac{2\sqrt{2}}{3-2}=2\sqrt{2} \notin \mathbb{Q}
\]
Иррациональное число
(д)
\[
\frac{\sqrt{2}+\sqrt{3}}{\sqrt{3}-\sqrt{2}}=\frac{(\sqrt{2}+\sqrt{3})^2}{(\sqrt{3}-\sqrt{2})(\sqrt{3}+\sqrt{2})}=\frac{2+2\sqrt{6}+3}{3-2}=5+2\sqrt{6} \notin \mathbb{Q}
\]
Иррациональное число
(е)
\[
\frac{\sqrt{5}}{\sqrt{5}-\sqrt{2}}+\frac{\sqrt{5}}{\sqrt{5}+\sqrt{2}}=\frac{\sqrt{5}(\sqrt{5}+\sqrt{2})+\sqrt{5}(\sqrt{5}-\sqrt{2})}{(\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2})}=\frac{5+\sqrt{10}+5-\sqrt{10}}{5-2}=\frac{10}{3}=3\frac{1}{3} \in \mathbb{Q}
\]
Рациональное число

Конечно, вот как будет выглядеть ваш код LaTeX на сайте:
(a)
\[
\frac{13}{7} \approx 1,8571 \ldots
\]
\[
\begin{aligned}
1,8<\frac{13}{7}<1,9, \quad \frac{13}{7} & \approx 1,8, \quad \frac{13}{7} \approx 1,9 \\
1,85<\frac{13}{7}<1,86, \quad \frac{13}{7} & \approx 1,85, \quad \frac{13}{7} \approx 1,86 \\
1,857<\frac{13}{7}<1,858, \quad \frac{13}{7} & \approx 1,857, \quad \frac{13}{7} \approx 1,858
\end{aligned}
\]
(б)
\[
\begin{gathered}
-\frac{13}{7} \approx-1,8571 \ldots \\
-1,9<-\frac{13}{7}<-1,8, \quad-\frac{13}{7} \stackrel{\downarrow}{\approx}-1,9, \quad-\frac{13}{7} \stackrel{\uparrow}{\approx}-1,8 \\
-1,86<-\frac{13}{7}<-1,85, \quad-\frac{13}{7} \stackrel{\downarrow}{\approx}-1,86,-\frac{13}{7} \stackrel{\uparrow}{\approx}-1,85 \\
-1,858<-\frac{13}{7}<-1,857,-\frac{13}{7} \stackrel{\downarrow}{\approx}-1,858,-\frac{13}{7} \stackrel{\uparrow}{\approx}-1,857
\end{gathered}
\]
(в)
\[
\begin{gathered}
\frac{5}{16}=0,3125 \\
0,3<\frac{5}{16}<0,4, \quad \frac{5}{16} \stackrel{\downarrow}{\approx} 0,3, \quad \frac{5}{16} \stackrel{\uparrow}{\approx} 0,4 \\
0,31<\frac{5}{16}<0,32, \quad \frac{5}{16} \stackrel{\downarrow}{\approx} 0,31, \quad \frac{5}{16} \stackrel{\uparrow}{\approx} 0,32 \\
0,312<\frac{5}{16}<0,313, \quad \frac{5}{16} \stackrel{\downarrow}{\approx} 0,312, \quad \frac{5}{16} \stackrel{\uparrow}{\approx} 0,313
\end{gathered}
\]
(г)
\[
\begin{gathered}
-\frac{5}{16}=-0,3125 \\
-0,4<-\frac{5}{16}<-0,3, \quad-\frac{5}{16} \stackrel{\downarrow}{\approx}-0,4,-\frac{5}{16} \stackrel{\uparrow}{\approx}-0,3 \\
-0,32<-\frac{5}{16}<-0,31, \quad-\frac{5}{16} \stackrel{\downarrow}{\approx}-0,32,-\frac{5}{16} \stackrel{\uparrow}{\approx}-0,31 \\
-0,313<-\frac{5}{16}<-0,312, \quad-\frac{5}{16} \stackrel{\downarrow}{\approx}-0,313,-\frac{5}{16} \stackrel{\uparrow}{\approx}-0,312
\end{gathered}
\]
(д)
\[
\begin{aligned}
& \sqrt{3} \approx 1,7320 \ldots \\
& 1,7<\sqrt{3}<1,8, \quad \sqrt{3} \stackrel{\downarrow}{\approx} 1,7, \quad \sqrt{3} \stackrel{\uparrow}{\approx} 1,8 \\
& 1,73<\sqrt{3}<1,74, \quad \sqrt{3} \stackrel{\downarrow}{\approx} 1,73, \quad \sqrt{3} \stackrel{\uparrow}{\approx} 1,74 \\
& 1,732<\sqrt{3}<1,733, \quad \sqrt{3} \stackrel{\downarrow}{\approx} 1,732, \quad \sqrt{3} \stackrel{\uparrow}{\approx} 1,733 \\
&
\end{aligned}
\]
(е)
\[
\begin{gathered}
-\sqrt{3} \approx-1,7320 \ldots \\
-1,8<-\sqrt{3}<-1,7, \quad-\sqrt{3} \stackrel{\downarrow}{\approx}-1,8,-\sqrt{3} \stackrel{\uparrow}{\approx}-1,7 \\
-1,74<-\sqrt{3}<-1,73, \quad-\sqrt{3} \stackrel{\downarrow}{\approx}-1,74,-\sqrt{3} \stackrel{\uparrow}{\approx}-1,73 \\
-1,733<-\sqrt{3}<-1,732, \quad-\sqrt{3} \stackrel{\downarrow}{\approx
}-1,733,-\sqrt{3} \stackrel{\uparrow}{\approx}-1,732
\end{gathered}
\]