Numeri razionali e reali 303 e 305 da incubo.

Un solo esercizio per volta: leggere il REGOLAMENTO

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Ti faccio la 305:

$\small \left\{\left[\left(-\dfrac{3}{5}\right)^6·\left(\dfrac{25}{9}\right)^2\right]^4 : \left(\dfrac{9}{5}\right)^6·27^2\right\} : \left(-\dfrac{3}{25}\right)=$

$\small =\left\{\left[-\dfrac{3^6}{5^6}·\dfrac{25^2}{9^2}\right]^4·\left(\dfrac{5}{9}\right)^6·27^2\right\}·\left(-\dfrac{25}{3}\right)=$

$\small =\left\{\left[-\dfrac{3^6}{5^6}·\dfrac{5^2·5^2}{3^2·3^2}\right]^4·\dfrac{5^6}{9^6}·9^2·3^2\right\}·\left(-\dfrac{25}{3}\right)=$

$\small =\left\{\left[3^{6-2-2}·5^{2+2-6}\right]^4·\dfrac{5^6·9^2·3^2}{9^6}\right\}·\left(-\dfrac{25}{3}\right)=$

$\small =\left\{\left[3^2·5^{-2}\right]^4·5^6·9^{2-6}·3^2\right\}·\left(-\dfrac{25}{3}\right)=$

$\small =\left\{\left[3^2·\left(\dfrac{1}{5}\right)^2\right]^4·5^6·9^{-4}·3^2\right\}·\left(-\dfrac{25}{3}\right)=$

$\small =\left\{\left[\dfrac{3^2}{5^2}\right]^4·5^6·\left(\dfrac{1}{9}\right)^4·3^2\right\}·\left(-\dfrac{25}{3}\right)=$

$\small =\left\{\dfrac{3^{2·4}}{5^{2·4}}·5^6·\dfrac{1^4}{3^4·3^4}·3^2\right\}·\left(-\dfrac{25}{3}\right)=$

$\small =\left\{\dfrac{3^8}{5^8}·5^6·\dfrac{1}{3^4·3^4}·3^2\right\}·\left(-\dfrac{25}{3}\right)=$

$\small =\left\{3^{8-4-4+2}·5^{6-8}\right\}·\left(-\dfrac{25}{3}\right)=$

$\small =\left\{3^2·5^{-2}\right\}·\left(-\dfrac{25}{3}\right)=$

$\small =\left\{3^2·\dfrac{1}{5^2}\right\}·\left(-\dfrac{25}{3}\right)=$

$\small = \dfrac{3^2}{5^2}·\left(-\dfrac{25}{3}\right)=$

$\small = \dfrac{\cancel9^3}{\cancel{25}_1}·\left(-\dfrac{\cancel{25}^1}{\cancel3_1}\right)=$

$\small = \dfrac{3}{1}·-\dfrac{1}{1}= -3$