
Pour cet exercice, il faut se souvenir des identités remarquables:
\[
(a+b)^2=a^2+2ab+b^2\qquad\text{et}\qquad (a-b)^2=a^2-2ab+b^2.
\]
\begin{enumerate}
\item $\begin{aligned}[t]
\big(5+\sqrt3\big)^2 & = 5^2 + 2\times5\times\sqrt3 + \big(\sqrt3\big)^2\\
& = 25 + 10\sqrt3 + 3\\
& = \underline{28 + 10\sqrt3}.
\end{aligned}$

\item $\begin{aligned}[t]
\big(7-\sqrt2\big)^2 & = 7^2 - 2\times7\times\sqrt2 + \big(\sqrt2\big)^2\\
& = 49 - 14\sqrt2 + 2\\
& = \underline{51 - 14\sqrt2}.
\end{aligned}$

\item $\begin{aligned}[t]
\big(-1+2\sqrt7\big)^2 & = (-1)^2 + 2\times(-1)\times2\sqrt7 + \big(2\sqrt7\big)^2\\
& = 1 - 4\sqrt7 + 2^2\big(\sqrt7\big)^2\\
& = 1 - 4\sqrt7 + 4\times7\\
& = 1 - 4\sqrt7 + 28\\
& = \underline{29 - 4\sqrt7}
\end{aligned}$

\item $\begin{aligned}[t]
\big(-3-4\sqrt2\big)^2 & = \big[(-1)(3+4\sqrt2)\big]^2\\
& = (-1)^2\big(3+4\sqrt2\big)^2\\
& = \big(3+4\sqrt2\big)^2\\
& = 3^2 + 2\times3\times4\sqrt2 + \big(4\sqrt2\big)^2\\
& = 9 + 24\sqrt2 + 4^2\big(\sqrt2)^2\\
& = 9 + 24\sqrt2+16\times2
\end{aligned}$

$\begin{aligned}[t]
& = 9 + 24\sqrt2 + 32\\
& = \underline{41 + 24\sqrt2}.
\end{aligned}$
\end{enumerate}
