Operatii aritmetice, radicali

Natalia Lun Apr 09, 2012 7:05 pm

Am 3 exercitii la care m-am impotmolit tare:

1. $\sqrt2{^{4}}+ \sqrt{\frac{5}{0.0(2)}}+\sqrt{\frac{55}{0.0(02)}}+\sqrt{\frac{555}{0.0(002)}}=$
2. $(\sqrt{2}+1)^{2}+\sqrt{2}(3\sqrt{2}-2)+( \sqrt{17}-1)(\sqrt{17}+1 )$
3. $324*253+ (1+2+3+...+647)=$

Multumesc!!! cheers

Natalia Lun Apr 09, 2012 7:06 pm

la primul exercitiu puterea e sub radical

**Admin** Lun Apr 09, 2012 10:08 pm

Natalia a scris:
2. $(\sqrt{2}+1)^{2}+\sqrt{2}(3\sqrt{2}-2)+( \sqrt{17}-1)(\sqrt{17}+1 )$

Soluție :
$\begin{array}{l} E = {\left( {\sqrt 2 + 1} \right)^2} + \sqrt 2 \left( {3\sqrt 2 - 2} \right) + \left( {\sqrt {17} - 1} \right)\left( {\sqrt {17} + 1} \right)\\ \left\{ \begin{array}{l} {\left( {\sqrt 2 + 1} \right)^2} = 2 + 2\sqrt 2 + 1 = 3 + 2\sqrt 2 \\ \sqrt 2 \left( {3\sqrt 2 - 2} \right) = 3 \times \sqrt 2 \times \sqrt 2 - 2\sqrt 2 = 6 - 2\sqrt 2 \\ \left( {\sqrt {17} - 1} \right)\left( {\sqrt {17} + 1} \right) = {\left( {\sqrt {17} } \right)^2} - 1 = 17 - 1 = 16 \end{array} \right. \Rightarrow \\ E = 3 + 2\sqrt 2 + 6 - 2\sqrt 2 + 16 = 25 \end{array}$

**Admin** Lun Apr 09, 2012 10:15 pm

Natalia a scris:
3. $324*253+ (1+2+3+...+647)=$

$\begin{gathered} E = 324 \times 253 + {\rm{ }}\left( {1 + 2 + 3 + ... + 647} \right) \hfill \\ 1 + 2 + 3 + ... + 647 = ? \hfill \\ {\text{Folosim formula lui Gauss }}: \hfill \\ 1 + 2 + 3 + ... + 647 = \frac{{647 \times 648}}{2} = 209628 \hfill \\ E = 324 \times 253 + 209628 = 291600 \hfill \\ \end{gathered}$

Natalia Mar Apr 10, 2012 3:32 pm

Multumesc! Si la primul exercitiu?

**Admin** Mar Apr 10, 2012 11:32 pm

Natalia a scris:
1. $\sqrt2{^{4}}+ \sqrt{\frac{5}{0.0(2)}}+\sqrt{\frac{55}{0.0(02)}}+\sqrt{\frac{555}{0.0(002)}}=$

$\begin{gathered} E = \sqrt {{2^4}} + \sqrt {\frac{5}{{0.0(2)}}} + \sqrt {\frac{{55}}{{0.0(02)}}} + \sqrt {\frac{{555}}{{0.0(002)}}} \hfill \\ \sqrt {{2^4}} = \sqrt {{4^2}} = 4 \hfill \\ \left\{ \begin{gathered} \sqrt {\frac{5}{{0.0(2)}}} = \sqrt {\frac{5}{{\frac{1}{{45}}}}} = \sqrt {5 \times 45} = \sqrt {225} = 15 \hfill \\ 0.0(2) = \frac{2}{{90}} = \frac{1}{{45}} \hfill \\ \end{gathered} \right. \hfill \\ \left\{ \begin{gathered} \sqrt {\frac{{55}}{{0.0(02)}}} = \sqrt {\frac{{55}}{{\frac{1}{{495}}}}} = \sqrt {55 \times 495} = 165 \hfill \\ 0.0(02) = \frac{2}{{990}} = \frac{1}{{495}} \hfill \\ \end{gathered} \right. \hfill \\ \left\{ \begin{gathered} \sqrt {\frac{{555}}{{0.0(002)}}} = \sqrt {\frac{{555}}{{\frac{1}{{4995}}}}} = \sqrt {555 \times 4995} = 1665 \hfill \\ 0.0(002) = \frac{2}{{9990}} = \frac{1}{{4995}} \hfill \\ \end{gathered} \right. \hfill \\ E = 4 + 15 + 165 + 1665 = .... \hfill \\ \end{gathered}$

Natalia Mier Apr 11, 2012 8:53 pm

Aaa. Multumesc mult mult de tot!!!!! cheers

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Operatii aritmetice, radicali

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