Aplicăm:

• dacă \({a, b, x, y \in ℝ}\) și \({x, y \ge 0}\), atunci:



\({a\sqrt{x} :b\sqrt{y} = (a:b)\sqrt{x:y}}\)



• regula semnelor:



\({\textbf{+} \; \textbf{:} \; \textbf{+} \; \textbf{=} \; \textbf{–} \; \textbf{:} \; \textbf{–} \; \textbf{=} \; \textbf{+}}\)




\({\textbf{–} \; \textbf{:} \; \textbf{+} \; \textbf{=} \; \textbf{+} \; \textbf{:} \; \textbf{–} \; \textbf{=} \; \textbf{–}}\)

a) \({\sqrt{6} :\sqrt{3} = \sqrt{6:3}=\sqrt{2}}\)

b) \({\sqrt{12} :\sqrt{6} = \sqrt{12:6}=\sqrt{2}}\)

c) \({(-\sqrt{8}) :\sqrt{2} =[(-1):1] \cdot \sqrt{8:2}=-\sqrt{4}}\)

d) \({(-10\sqrt{12}) :(-\sqrt{6}) = [(-10):(-1)] \cdot \sqrt{12 : 6}=10\sqrt{2}}\)

e) \({6\sqrt{10}:(-3\sqrt{2}) =[6:(-3)] \cdot \sqrt{10 : 2}=-2\sqrt{5}}\)

f) \({(-15\sqrt{18}):5\sqrt{6} =[(-15) : 5]\cdot \sqrt{18 : 6} =-3\sqrt{3}}\)

g) \({21\sqrt{12}:(-3\sqrt{3}) = [21:(-3)] \cdot \sqrt{12 : 3}=-7\sqrt{4}=(-7) \cdot 2=-14 }\)

h) \({(-36\sqrt{14}) :(-9\sqrt{7}) = [(-36) : (-9)] \cdot \sqrt{14 : 7}=4\sqrt{2}}\)

i) \({25\sqrt{45} :5\sqrt{5} = (25:5)\sqrt{45 : 5}=5\sqrt{9}=5 \cdot 3=15}\)

j) \({48\sqrt{21} :(-12) = [48:(-12)] \cdot\sqrt{21}=-4\sqrt{21}}\)

k) \({(-48\sqrt{54}) : 4\sqrt{6} = [(-48):4]\sqrt{54:6}=-12\sqrt{9}=-12 \cdot 3=-36}\)

l) \({6\sqrt{28} : (-3\sqrt{7}) = [6:(-3)]\sqrt{28:7}=-2\sqrt{4}=(-2 ) \cdot 2 = -4}\)

m) \({10\sqrt{45} : (-5\sqrt{3}) = [10:(-5)]\sqrt{45:3}=-2\sqrt{15}}\)

n) \({22\sqrt{300} : (-\sqrt{10}) = [22:(-1)]\sqrt{300:10}=-22\sqrt{30}}\)

a) \({\frac{\displaystyle \sqrt{10}}{\displaystyle \sqrt{5}} = \sqrt{10:5}=\sqrt{2}}\)

b) \({\frac{\displaystyle \sqrt{15}}{\displaystyle \sqrt{3}} = \sqrt{15:3}=\sqrt{5}}\)

c) \({\frac{\displaystyle 8\sqrt{20}}{\displaystyle 4\sqrt{10}} =(8:4)\sqrt{20:10}=2\sqrt{2}}\)

d) \({-\frac{\displaystyle 16\sqrt{6}}{\displaystyle 2\sqrt{3}} =-(16:2)\sqrt{6:3}=-8\sqrt{2}}\)

Aplicăm:

• dacă \({a, b \in ℝ}\) și \({b \neq 0}\), atunci:



\({a :b=a \cdot \frac{\displaystyle 1}{\displaystyle b}}\)



• dacă \({a, b, c, d \in ℝ}\), \({a, b, c, d \ge 0}\) și \({b, c,d \neq 0}\), atunci:



\({\frac{\displaystyle \sqrt{a}}{\displaystyle \sqrt{b}} : \frac{\displaystyle \sqrt{c}}{\displaystyle \sqrt{d}} = \frac{\displaystyle \sqrt{a}}{\displaystyle \sqrt{b}} \cdot \frac{\displaystyle \sqrt{d}}{\displaystyle \sqrt{c}}}\)

\({\textcolor{white}{\frac{\displaystyle \sqrt{a}}{\displaystyle \sqrt{b}} : \frac{\displaystyle \sqrt{c}}{\displaystyle \sqrt{d}}} = \sqrt{\frac{\displaystyle a \cdot d}{\displaystyle b \cdot c}}}\)

a) \({5\sqrt{5} : \frac{\displaystyle 10}{\displaystyle \sqrt{5}} = \cancel{5}\sqrt{5} \cdot \frac{\displaystyle \sqrt{5}}{\displaystyle \cancel{10}_2}}\)

a) \({\textcolor{white}{5\sqrt{5} : \frac{\displaystyle 10}{\displaystyle \sqrt{5}}} =\frac{\displaystyle \sqrt{5} \cdot \sqrt{5} }{\displaystyle 2}}\)

a) \({\textcolor{white}{5\sqrt{5} : \frac{\displaystyle 10}{\displaystyle \sqrt{5}}} =\frac{\displaystyle \sqrt{5 \cdot 5}}{\displaystyle 2}}\)

a) \({\textcolor{white}{5\sqrt{5} : \frac{\displaystyle 10}{\displaystyle \sqrt{5}}}=\frac{\displaystyle \sqrt{5^2}}{\displaystyle 2}}\)

a) \({\textcolor{white}{5\sqrt{5} : \frac{\displaystyle 10}{\displaystyle \sqrt{5}}} =\frac{\displaystyle 5}{\displaystyle 2}}\)

b) \({6\sqrt{7} : \frac{\displaystyle 14}{\displaystyle \sqrt{42}}= \cancel{6}^3\sqrt{7} \cdot \frac{\displaystyle \sqrt{42}}{\displaystyle \cancel{14}^7}}\)

b) \({ \textcolor{white}{6\sqrt{7} : \frac{\displaystyle 14}{\displaystyle \sqrt{42}}}= \frac{\displaystyle 3\sqrt{7} \cdot\sqrt{42}}{\displaystyle 7}}\)

b) \({ \textcolor{white}{6\sqrt{7} : \frac{\displaystyle 14}{\displaystyle \sqrt{42}}}= \frac{\displaystyle 3\sqrt{7 \cdot 42}}{\displaystyle 7}}\)

b) \({ \textcolor{white}{6\sqrt{7} : \frac{\displaystyle 14}{\displaystyle \sqrt{42}}}= \frac{\displaystyle 3\sqrt{7 \cdot 7 \cdot 6}}{\displaystyle 7}}\)

b) \({ \textcolor{white}{6\sqrt{7} : \frac{\displaystyle 14}{\displaystyle \sqrt{42}}}= \frac{\displaystyle 3\sqrt{7^2 \cdot 6}}{\displaystyle 7}}\)

b) \({ \textcolor{white}{6\sqrt{7} : \frac{\displaystyle 14}{\displaystyle \sqrt{42}}}= \frac{\displaystyle 3 \cdot \cancel{7} \cdot \sqrt{6}}{\displaystyle \cancel{7}}}\)

b) \({ \textcolor{white}{6\sqrt{7} : \frac{\displaystyle 14}{\displaystyle \sqrt{42}}}= 3\sqrt{6}}\)

c) \({6\sqrt{60} : \frac{\displaystyle 5}{\displaystyle 2\sqrt{15}} = 6\sqrt{60} \cdot \frac{\displaystyle 2\sqrt{15}}{\displaystyle 5}}\)

c) \({\textcolor{white}{6\sqrt{60} : \frac{\displaystyle 5}{\displaystyle 2\sqrt{15}} }= \frac{\displaystyle 6\sqrt{60} \cdot 2\sqrt{15}}{\displaystyle 5}}\)

c) \({\textcolor{white}{6\sqrt{60} : \frac{\displaystyle 5}{\displaystyle 2\sqrt{15}} }= \frac{\displaystyle (6 \cdot 2) \cdot \sqrt{60 \cdot 15}}{\displaystyle 5}}\)

c) \({\textcolor{white}{6\sqrt{60} : \frac{\displaystyle 5}{\displaystyle 2\sqrt{15}} }= \frac{\displaystyle 12 \sqrt{4 \cdot 15 \cdot 15}}{\displaystyle 5}}\)

c) \({\textcolor{white}{6\sqrt{60} : \frac{\displaystyle 5}{\displaystyle 2\sqrt{15}} }= \frac{\displaystyle 12 \sqrt{2^2 \cdot 15^2}}{\displaystyle 5}}\)

c) \({\textcolor{white}{6\sqrt{60} : \frac{\displaystyle 5}{\displaystyle 2\sqrt{15}} }= \frac{\displaystyle 12 \cdot 2 \cdot \cancel{15}^3 }{\displaystyle \cancel{5} }}\)

c) \({\textcolor{white}{6\sqrt{60} : \frac{\displaystyle 5}{\displaystyle 2\sqrt{15}} }= 12 \cdot 2 \cdot 3}\)

c) \({\textcolor{white}{6\sqrt{60} : \frac{\displaystyle 5}{\displaystyle 2\sqrt{15}} }= 72}\)

d) \({\sqrt{294} : \frac{\displaystyle 28}{\displaystyle 8\sqrt{3}} = \sqrt{294} \cdot \frac{\displaystyle 8\sqrt{3}}{\displaystyle 28} }\)

d) \({\textcolor{white}{\sqrt{294} : \frac{\displaystyle 28}{\displaystyle 8\sqrt{3}} }= \frac{\displaystyle \sqrt{294} \cdot 8\sqrt{3}}{\displaystyle 28} }\)

d) \({\textcolor{white}{\sqrt{294} : \frac{\displaystyle 28}{\displaystyle 8\sqrt{3}} }= \frac{\displaystyle \cancel{8}^2\sqrt{294 \cdot 3}}{\displaystyle \cancel{28}^7} }\)

d) \({\textcolor{white}{\sqrt{294} : \frac{\displaystyle 28}{\displaystyle 8\sqrt{3}} }= \frac{\displaystyle 2\sqrt{2 \cdot 3 \cdot 7^2 \cdot 3}}{\displaystyle 7} }\)

d) \({\textcolor{white}{\sqrt{294} : \frac{\displaystyle 28}{\displaystyle 8\sqrt{3}} }= \frac{\displaystyle 2\sqrt{2 \cdot 3^2 \cdot 7^2 }}{\displaystyle 7} }\)

d) \({\textcolor{white}{\sqrt{294} : \frac{\displaystyle 28}{\displaystyle 8\sqrt{3}} }= \frac{\displaystyle 2 \cdot 3 \cdot \cancel{7} \cdot \sqrt{2 }}{\displaystyle \cancel{7}} }\)

d) \({\textcolor{white}{\sqrt{294} : \frac{\displaystyle 28}{\displaystyle 8\sqrt{3}} }= 6 \sqrt{2 }}\)

Am descompus în factori primi numărul 294, astfel:

e) \({\sqrt{567} : \frac{\displaystyle 27}{\displaystyle \sqrt{21}} = \sqrt{567} \cdot \frac{\displaystyle \sqrt{21}}{\displaystyle 27}}\)

e) \({\textcolor{white}{\sqrt{567} : \frac{\displaystyle 27}{\displaystyle \sqrt{21}}} = \frac{\displaystyle \sqrt{567} \cdot \sqrt{21}}{\displaystyle 27}}\)

e) \({\textcolor{white}{\sqrt{567} : \frac{\displaystyle 27}{\displaystyle \sqrt{21}}} = \frac{\displaystyle \sqrt{567 \cdot 21}}{\displaystyle 27}}\)

e) \({\textcolor{white}{\sqrt{567} : \frac{\displaystyle 27}{\displaystyle \sqrt{21}}} = \frac{\displaystyle \sqrt{3^4 \cdot 7 \cdot 3 \cdot 7}}{\displaystyle 27}}\)

e) \({\textcolor{white}{\sqrt{567} : \frac{\displaystyle 27}{\displaystyle \sqrt{21}}} = \frac{\displaystyle \sqrt{3^4 \cdot 7^2 \cdot 3 }}{\displaystyle 27}}\)

e) \({\textcolor{white}{\sqrt{567} : \frac{\displaystyle 27}{\displaystyle \sqrt{21}}} = \frac{\displaystyle \cancel{3^2} \cdot 7 \cdot \sqrt{3 }}{\displaystyle \cancel{27}_3}}\)

e) \({\textcolor{white}{\sqrt{567} : \frac{\displaystyle 27}{\displaystyle \sqrt{21}}} = \frac{\displaystyle 7 \sqrt{3 }}{\displaystyle 3}}\)

Am descompus în factori primi numărul 567, astfel:

f) \({\frac{\displaystyle \sqrt{10}}{\displaystyle \sqrt{18}} : \frac{\displaystyle \sqrt{30}}{\displaystyle \sqrt{12}} = \frac{\displaystyle \sqrt{10}}{\displaystyle \sqrt{18}} \cdot \frac{\displaystyle \sqrt{12}}{\displaystyle \sqrt{30}}}\)

f) \({\textcolor{white}{\frac{\displaystyle \sqrt{10}}{\displaystyle \sqrt{18}} : \frac{\displaystyle \sqrt{30}}{\displaystyle \sqrt{12}}} = \sqrt{\frac{\displaystyle \cancel{10} \cdot \bcancel{12}^2}{\displaystyle \bcancel{18}_3 \cdot \cancel{30}_3}}}\)

f) \({\textcolor{white}{\frac{\displaystyle \sqrt{10}}{\displaystyle \sqrt{18}} : \frac{\displaystyle \sqrt{30}}{\displaystyle \sqrt{12}}} = \sqrt{\frac{\displaystyle 2}{\displaystyle 3 \cdot 3}}}\)

f) \({\textcolor{white}{\frac{\displaystyle \sqrt{10}}{\displaystyle \sqrt{18}} : \frac{\displaystyle \sqrt{30}}{\displaystyle \sqrt{12}}} = \sqrt{\frac{\displaystyle 2}{\displaystyle 9}}}\)

f) \({\textcolor{white}{\frac{\displaystyle \sqrt{10}}{\displaystyle \sqrt{18}} : \frac{\displaystyle \sqrt{30}}{\displaystyle \sqrt{12}}} = \frac{\displaystyle \sqrt{2}}{\displaystyle \sqrt{9}}}\)

f) \({\textcolor{white}{\frac{\displaystyle \sqrt{10}}{\displaystyle \sqrt{18}} : \frac{\displaystyle \sqrt{30}}{\displaystyle \sqrt{12}}} = \frac{\displaystyle \sqrt{2}}{\displaystyle \sqrt{3^2}}}\)

f) \({\textcolor{white}{\frac{\displaystyle \sqrt{10}}{\displaystyle \sqrt{18}} : \frac{\displaystyle \sqrt{30}}{\displaystyle \sqrt{12}}} = \frac{\displaystyle \sqrt{2}}{\displaystyle 3}}\)