Question

O valor numérico da expressão
$\mathsf{\left[\dfrac{\left(\dfrac{a}{c}\right)^2+2\cdot\left(\dfrac{1}{a}\right)\cdot\left(\dfrac{1}{b}\right)-\left(\dfrac{2}{d}\right)^2}{\left(\dfrac{a}{b}+\dfrac{d}{c}\right)^2}\right]}$
para a = 16; b = 14; c = 18 e d = 20 é:
(simplifique o resultado o máximo possível)​

Answer 1

Olá! obtenha a resposta ao decorrer dos cálculos.

Queremos descobrir o valor numérico da seguinte expressão:

$\orange{\boxed{\mathsf{\left[\dfrac{\left(\dfrac{a}{c}\right)^2+2\cdot\left(\dfrac{1}{a}\right)\cdot\left(\dfrac{1}{b}\right)-\left(\dfrac{2}{d}\right)^2}{\left(\dfrac{a}{b}+\dfrac{d}{c}\right)^2}\right]}}}$

Para:

$\pink{\boxed{\boxed{\begin{array}{l}\\\orange{\tt a=16;~~b=14;~~c=18;~~d=20}\\\\\end{array}}}}$

Vamos, resolver primeiro a parte de cima (Numerador) da expressão:

$\green{\mathsf{\left(\dfrac{a}{c}\right)^2+2\cdot\left(\dfrac{1}{a}\right)\cdot\left(\dfrac{1}{b}\right)-\left(\dfrac{2}{d}\right)^2}}$

Substituindo os valores das letras, obtemos:

$\mathsf{\implies \blue{\left(\dfrac{16}{18}\right)^2+2\cdot \left(\dfrac{ 1}{16}\right)\cdot\left(\dfrac{1}{14}\right)-\left(\dfrac{2}{20}\right)^2}}$

$\mathsf{\iff\blue{ \left(\dfrac{8}{9}\right)^2+\left(\dfrac{1}{8}\right)\cdot\left(\dfrac{1}{14}\right)-\left(\dfrac{1}{10}\right)^2}}$

$\mathsf{\iff\blue{\left(\dfrac{64}{81}\right)+\left(\dfrac{1}{112}\right)-\left(\dfrac{1}{100} \right)}}$

$\mathsf{\iff\boxed{\blue{\sf \dfrac{ 17 8957}{ 226800}}}}$

Agora, vamos resolver a parte de baixo (Denominador) da expressão:

$\mathsf{ \green{\left(\dfrac{a}{b}+\dfrac{d}{c}\right)^2}}$

Substituindo os valores das letras, obtemos:

$\mathsf{\implies\blue{\left(\dfrac{16}{14}+\dfrac{20}{18}\right)^2 }}$

$\mathsf{\iff\blue{\left(\dfrac{142}{63}\right)^2 }}$

$\mathsf{\iff\boxed{\blue{\sf\dfrac{20164}{3969}}}}$

Por último vamos, simplificar o Numerador pelo Denominador:

$\mathsf{\implies\blue{\dfrac{178957}{226800}\div\dfrac{20164}{3969}}}$

$\mathsf{\iff\blue{\dfrac{17 8957}{226800}\cdot\dfrac{3969}{20164} }}$

$\mathsf{\iff\blue{ \dfrac{17 8957\cdot 3969}{226800\cdot 20164}}}$

$\mathsf{\iff\boxed{\blue{\sf \dfrac{1252699}{8065600} }}}$

Portanto, concluímos que:

$\red{\boxed{\boxed{\mathsf{\left[\dfrac{\left(\dfrac{a}{c}\right)^2+2\cdot\left(\dfrac{1}{a}\right)\cdot\left(\dfrac{1}{b}\right)-\left(\dfrac{2}{d}\right)^2}{\left(\dfrac{a}{b}+\dfrac{d}{c}\right)^2}\right]= \dfrac{1252699}{8065600}}} }}$

Logo, o resultado é 1252699/8065600.

$\boxed{~~\tt\red E\blue s\pink p\green e\orange r\purple o~\gray t \red e\blue r~\green a\orange j \gray u\pink d\blue a\purple d\red o\orange!\red!~~}$