Exercices sur les Racines Carrées

1) $A = (1+1)\sqrt{5} = 2\sqrt{5}$
$B = \dfrac{\sqrt{100 \times 5}}{5} = \dfrac{10\sqrt{5}}{5} = 2\sqrt{5}$
$C = 2 \times (\sqrt{5})^2 = 2 \times 5 = 10$
$D = \sqrt{4 \times 5} = 2\sqrt{5}$
$E = \sqrt{5} + 5$ (ne peut pas être simplifié)
Les trois nombres égaux sont $A$, $B$ et $D$ (tous égaux à $2\sqrt{5}$).

2) $F = 5 - 4 + 2\sqrt{5} = $ $1 + 2\sqrt{5}$ (avec $a=1, b=2$)
$G = \dfrac{1}{4} \times 28 - \dfrac{1}{4} \times 16\sqrt{5} = 7 - 4\sqrt{5} = $ $7 - 4\sqrt{5}$ (avec $a=7, b=-4$)

3) a) $M = 2\sqrt{6 \times 3} = 2\sqrt{18} = 2\sqrt{9 \times 2} = 2(3\sqrt{2}) = $ $6\sqrt{2}$
$N = \sqrt{16 \times 2} - 3\sqrt{25 \times 2} = 4\sqrt{2} - 3(5\sqrt{2}) = (4 - 15)\sqrt{2} = $ $-11\sqrt{2}$

3) b) $M \times N = (6\sqrt{2}) \times (-11\sqrt{2}) = 6 \times (-11) \times (\sqrt{2})^2 = -66 \times 2 = $ $-132$. C'est bien un entier relatif.

$I = 3\sqrt{4 \times 5} + \sqrt{9 \times 5} - \sqrt{36 \times 5} = 3(2\sqrt{5}) + 3\sqrt{5} - 6\sqrt{5} = (6 + 3 - 6)\sqrt{5} = $ $3\sqrt{5}$

$J = \sqrt{25 \times 3} - 2\sqrt{4 \times 3} + 2\sqrt{9 \times 3} = 5\sqrt{3} - 2(2\sqrt{3}) + 2(3\sqrt{3}) = (5 - 4 + 6)\sqrt{3} = $ $7\sqrt{3}$

$K = 2\sqrt{75 \times 6} = 2\sqrt{450} = 2\sqrt{225 \times 2} = 2(15\sqrt{2}) = $ $30\sqrt{2}$

$L = 4\sqrt{2 \times 90} = 4\sqrt{180} = 4\sqrt{36 \times 5} = 4(6\sqrt{5}) = $ $24\sqrt{5}$

1) $A = 3\sqrt{4 \times 5} + \sqrt{9 \times 5} = 3(2\sqrt{5}) + 3\sqrt{5} = (6 + 3)\sqrt{5} = $ $9\sqrt{5}$
$B = \sqrt{36 \times 5} - 3\sqrt{5} = 6\sqrt{5} - 3\sqrt{5} = (6 - 3)\sqrt{5} = $ $3\sqrt{5}$

2) $A \times B = (9\sqrt{5}) \times (3\sqrt{5}) = 9 \times 3 \times (\sqrt{5})^2 = 27 \times 5 = $ $135$. C'est bien un entier.
$\dfrac{A}{B} = \dfrac{9\sqrt{5}}{3\sqrt{5}} = \dfrac{9}{3} = $ $3$. C'est bien un entier.

$a = \sqrt{36} = $ $6$

$b = (64 \times 5) - (25 \times 5) = 320 - 125 = $ $195$

$c = (36 \times 5) - (4 + 5) = 180 - 9 = $ $171$

$d = \sqrt{\dfrac{32}{18}} = \sqrt{\dfrac{16}{9}} = $ $\dfrac{4}{3}$

$e = \sqrt{\dfrac{45 \times 22 \times 18}{20 \times 11}} = \sqrt{\dfrac{(9 \times 5) \times (2 \times 11) \times (2 \times 9)}{(4 \times 5) \times 11}} = \sqrt{\dfrac{81 \times 4 \times 5 \times 11}{4 \times 5 \times 11}} = \sqrt{81} = $ $9$

$f = \sqrt{\dfrac{3 \times 270}{10 \times 64}} = \sqrt{\dfrac{3 \times 27}{64}} = \sqrt{\dfrac{81}{64}} = $ $\dfrac{9}{8}$

$g = \sqrt{\dfrac{35 \times 7}{4 \times 45}} = \sqrt{\dfrac{(7 \times 5) \times 7}{4 \times (9 \times 5)}} = \sqrt{\dfrac{49}{36}} = $ $\dfrac{7}{6}$

$h = \sqrt{\dfrac{49}{100} + \dfrac{25}{10}} = \sqrt{\dfrac{49}{100} + \dfrac{250}{100}} = \sqrt{\dfrac{299}{100}} = $ $\dfrac{\sqrt{299}}{10}$

$i = \sqrt{\dfrac{4900}{\sqrt{300} \times \sqrt{12 \times 10^6}}} = \sqrt{\dfrac{4900}{(10\sqrt{3}) \times (1000\sqrt{12})}} = \sqrt{\dfrac{4900}{10000 \times \sqrt{36}}} = \sqrt{\dfrac{4900}{10000 \times 6}} = \sqrt{\dfrac{49}{600}} = \dfrac{7}{10\sqrt{6}} = $ $\dfrac{7\sqrt{6}}{60}$