Matematica

((1/2)^2 + (1/4)^2)/(1/2 + 1/4)^2=

=(1/4 + 1/16)/(3/4)^2=

=5/16/(9/16)= 5/9

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(1/2 + 1/3)/(1/2/1/3)/((1/2)^2 - (1/3)^2)=

=5/6/(1/2/1/3)/((1/4) - (1/9))=

=5/6/(1/2/1/3)/(5/36)=

=5/(5/36)= 36

125)

$\small  \dfrac{\left(\dfrac{1}{2}\right)^2+\left(\dfrac{1}{4}\right)^2}{\left(\dfrac{1}{2}\right)^2+\left(\dfrac{1}{4}\right)^2} =$

$\small = \dfrac{\dfrac{1}{4}+\dfrac{1}{16}}{\left(\dfrac{2+1}{4}\right)^2} =$

$\small = \dfrac{\dfrac{4+1}{16}}{\left(\dfrac{3}{4}\right)^2} =$

$\small = \dfrac{\dfrac{5}{16}}{\dfrac{9}{16}} = $

$\small = \dfrac{5}{\cancel{16}_1}·\dfrac{\cancel{16}^1}{9}= \dfrac{5}{9}$

1/4+1/16 = 5/16 

(1/2+1/4)^2 = (3/4)^2 = 9/16

5/16*16/9 = 5/9

1/2+1/3 = 5/6

1/2-1/3 = 1/6

5/6*6 = 5

(1/4-1/9) = (9-4)/36 = 5/36

3*36/5 = 36  

126)

$\small \dfrac{\dfrac{\frac{1}{2}+\frac{1}{3}}{\frac{1}{2}-\frac{1}{3}}}{\left(\frac{1}{2}\right)^2-\left(\frac{1}{3}\right)^2}=$

$\small =\dfrac{\dfrac{\frac{3+2}{6}}{\frac{3-2}{6}}}{\frac{1}{4}-\frac{1}{9}}=$

$\small =\dfrac{\dfrac{\frac{5}{6}}{\frac{1}{6}}}{\frac{9-4}{36}}=$

$\small =\dfrac{\dfrac{5}{6}·\dfrac{6}{1}}{\dfrac{5}{36}}=$

$\small =\dfrac{\dfrac{5}{\cancel6_1}·\dfrac{\cancel6^1}{1}}{\dfrac{5}{36}}=$

$\small =5·\dfrac{36}{5}=\cancel5^1·\dfrac{36}{\cancel5_1}= 36$