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1) \(\frac{\sqrt{165^2-124^2}}{164}=\frac{\sqrt{\left(165-124\right)\left(165+124\right)}}{164}=\frac{\sqrt{41}\cdot\sqrt{289}}{164}=\frac{\sqrt{41}\cdot17}{164}=\frac{17}{4\sqrt{41}}\)
2) \(\frac{\sqrt{149^2-76^2}}{\sqrt{457^2-384^2}}=\frac{\sqrt{\left(149+76\right)\left(149-76\right)}}{\sqrt{\left(457+384\right)\left(457-384\right)}}=\frac{\sqrt{225}\cdot\sqrt{73}}{\sqrt{841}\cdot\sqrt{73}}=\frac{25}{29}\)
a) \(\sqrt{\frac{\left(165-124\right)\left(165+124\right)}{164}}=\sqrt{\frac{41.289}{164}}=\sqrt{\frac{289}{4}}=\frac{17}{2}\)
b) tương tự ý a
c) \(\left(\sqrt{7+4\sqrt{3}}-\sqrt{7-4\sqrt{3}}\right)^2=7+4\sqrt{3}+7-4\sqrt{3}-2.\sqrt{7+4\sqrt{3}}.\sqrt{7-4\sqrt{3}}\)
\(=14-2\sqrt{\left(7+4\sqrt{3}\right)\left(7-4\sqrt{3}\right)}\)
\(=14-2\sqrt{49-48}\)
\(=14-2.1=12\)
\(\Rightarrow\sqrt{7+4\sqrt{3}}-\sqrt{7-4\sqrt{3}}=\sqrt{12}=2\sqrt{3}\)
a: \(=\sqrt{\dfrac{25}{16}\cdot\dfrac{49}{9}\cdot\dfrac{1}{100}}=\dfrac{5}{4}\cdot\dfrac{7}{3}\cdot\dfrac{1}{10}=\dfrac{35}{120}=\dfrac{7}{24}\)
b: \(=\sqrt{1.44\cdot0.81}=1.2\cdot0.9=1.08\)
c: \(=\sqrt{\dfrac{\left(165-124\right)\left(165+124\right)}{164}}=\sqrt{\dfrac{1}{4}\cdot289}=\dfrac{17}{2}\)
d: \(=\sqrt{\dfrac{\left(149-76\right)\left(149+76\right)}{\left(457-384\right)\left(457+384\right)}}=\sqrt{\dfrac{225}{841}}=\dfrac{15}{29}\)
a) \(\sqrt{\frac{165^2-124^2}{164}}=\sqrt{\frac{\left(165-124\right)\left(165+124\right)}{164}}=\sqrt{\frac{41.289}{164}}\)
\(=\sqrt{\frac{11849}{164}}=\sqrt{72,25}=8,5\)
b)\(\sqrt{\frac{149^2-76^2}{457^2-384^2}}=\sqrt{\frac{\left(149-76\right)\left(149+76\right)}{\left(457-384\right)\left(457+384\right)}}\) \(=\sqrt{\frac{73.225}{73.841}}=\sqrt{\frac{225}{841}}=\sqrt{\frac{15^2}{29^2}}=\frac{15}{29}\)
c)\(\sqrt{7+4\sqrt{3}}-\sqrt{7-4\sqrt{3}}\) \(=\sqrt{2^2+3+2.2.\sqrt{3}}-\sqrt{2^2+3-2.2.\sqrt{3}}\)
\(=\sqrt{2^2+2.2.\sqrt{3}+\sqrt{3}^2}-\sqrt{2^2-2.2.\sqrt{3}+\sqrt{3}^2}\)
\(=\sqrt{\left(2+\sqrt{3}\right)^2}-\sqrt{\left(2-\sqrt{3}\right)^2}=\left(2+\sqrt{3}\right)-\left(2-\sqrt{3}\right)\)
\(=2+\sqrt{3}-2+\sqrt{3}=2\sqrt{3}\)
a. =\(\sqrt{20.72.4,9}=\sqrt{2.72.49}=\sqrt{144.49}=12.7=84\)
b. \(\sqrt{\frac{999}{111}}=\sqrt{9}=3\)
c. = \(\sqrt{9472+27256}=\sqrt{36728}\approx191,645\)
d. = \(\sqrt{\frac{\left(149+76\right)\left(149-76\right)}{\left(457+348\right)\left(457-348\right)}}=\sqrt{\frac{225.73}{805.109}}=\sqrt{\frac{3285}{17549}}\approx136,817\)
\(\frac{1}{2}\left(x+y+z+457\right)=4\sqrt{x-4}+3\sqrt{y-2}+35\sqrt{z-787}\)
\(\Leftrightarrow x+y+z+457=8\sqrt{x-4}+6\sqrt{y-2}+70\sqrt{z-787}\)
\(\Leftrightarrow\left(x-4-8\sqrt{x-4}+16\right)+\left(y-2-6\sqrt{y-2}+9\right)+\left(z-787-70\sqrt{z-787}+1225\right)=0\)
\(\Leftrightarrow\left(\sqrt{x-4}-4\right)^2+\left(\sqrt{y-2}-3\right)^2+\left(\sqrt{z-787}-35\right)^2=0\)
\(\Leftrightarrow x=20;y=11;z=2012\)
