Les racines carrées et leurs propriétés (3)

Exercice 1

• $\sqrt{45}=\sqrt{9\times5}=\sqrt{9}\sqrt{5}=3\sqrt{5}$
• $\sqrt{24}=\sqrt{4\times6}=\sqrt{4}\sqrt{6}=2\sqrt{6}$
• $5\sqrt{2}=\sqrt{25}\sqrt{2}=\sqrt{25\times2}=\sqrt{50}$

Exercice 2

A = $\sqrt{8}\times\sqrt{2}=\sqrt{8\times2}=\sqrt{16}=4$
B = $3\sqrt{2}\times5\sqrt{14}\times\sqrt{7}=3\times5\times\sqrt{2\times14\times7}=15\sqrt{196}=15\times14=210$
C = $\dfrac{3\sqrt{72}}{5\sqrt{2}}=\dfrac{3}{5}\times\sqrt{\dfrac{72}{2}}=\dfrac{3}{5}\times\sqrt{36}=\dfrac{3}{5}\times6=\dfrac{18}{5}$
D = $\sqrt{2}\times\sqrt{3}=\sqrt{6}$
E = $7\sqrt{3}-2\sqrt{3}=5\sqrt{3}$
F = $\sqrt{50}-\sqrt{18}+\sqrt{72}=\sqrt{25\times2}-\sqrt{9\times2}+\sqrt{36\times2}=5\sqrt{2}-3\sqrt{2}+6\sqrt{2}=8\sqrt{2}$
G = $\sqrt{75}+2\sqrt{27}-\sqrt{12}=\sqrt{25\times3}+2\sqrt{9\times3}-\sqrt{4\times3}=5\sqrt{3}+6\sqrt{3}-2\sqrt{3}=9\sqrt{3}$
H = $\sqrt{3}\times\sqrt{12}=\sqrt{36}=6$
I = $\sqrt{2}\times\sqrt{0,02}=\sqrt{0,04}=\sqrt{4\times10^{-2}}=\sqrt{4}\times\sqrt{10^{-2}}=2\times10^{-1}=0,2$
J = $\sqrt{3}\times2\sqrt{3}\times2\sqrt{7}\times5\sqrt{7}=2\sqrt{3}^2\times2\times5\sqrt{7}^2=2\times3\times10\times7=420$
K = $\sqrt{3}\times\sqrt{27}=\sqrt{3}\times\sqrt{9\times3}=3\sqrt{3}^2=9$
L = $\sqrt{\dfrac{1}{7}}\times\sqrt{63}=\sqrt{\dfrac{63}{7}}=\sqrt{9}=3$
M = $\dfrac{\sqrt{7}}{\sqrt{6}}\times\dfrac{\sqrt{2}}{\sqrt{14}}\times\dfrac{\sqrt{21}}{\sqrt{18}}\times\dfrac{\sqrt{36}}{\sqrt{7}}=\sqrt{\dfrac{7\times2\times21\times36}{6\times14\times18\times7}}=1$
N = $\sqrt{9}+\sqrt{4}+\sqrt{25}=3+2+5=10$
O = $\sqrt{64+36}=\sqrt{100}=10$

Exercice 3

$A=x^2-2$
En remarquant que $2=(\sqrt{2})^2$, on reconnaît $a^2-b^2=(a-b)(a+b)$
$A=(x-\sqrt{2})(x+\sqrt{2})$

$B=4x^2-5$
En remarquant que $5=(\sqrt{5})^2$, on reconnaît $a^2-b^2=(a-b)(a+b)$
$B=(2x-\sqrt{5})(2x+\sqrt{5})$

Exercice 4

$\sqrt{25}=5$
$\sqrt{0}=0$
$\sqrt{1}=1$
$\sqrt{7^2}=7$
$3\sqrt{81}=27$
$(\sqrt{5})^2=5$
$(3\sqrt{2})^2=18$
$(-\sqrt{3})^2=3$
$(-\sqrt{5})^4=25$
$\sqrt{(-2)^6}=8$
$\sqrt{a^6}=a^3$

Exercice 5

A = $(\sqrt{7}-3)(\sqrt{7}+3)=\sqrt{7}^2-3^2=7-9=-2$

B = $(\sqrt{5}+\sqrt{2})^2=\sqrt{5}^2+2\sqrt{5}\sqrt{2}+\sqrt{2}^2=5+2\sqrt{10}+2=7+2\sqrt{10}$

C = $(\sqrt{8}+\sqrt{2})^2=\sqrt{8}^2+2\sqrt{8}\sqrt{2}+\sqrt{2}^2=8+2\sqrt{16}+2=10+8=18$

$D = (3\sqrt{2}+2\sqrt{3})^2=(3\sqrt{2})^2+2\times3\sqrt{2}\times2\sqrt{3}+(2\sqrt{3})^2=18+12\sqrt{6}+12=30+12\sqrt{6}$

E = $(\sqrt{2}-5)(\sqrt{2}+5)=\sqrt{2}^2-5^2=2-25=-23$

F = $(2\sqrt{3}+1)^2=(2\sqrt{3})^2+2\times2\sqrt{3}\times1+1^2=12+4\sqrt{3}+1=13+4\sqrt{3}$