Chiedo aiuto per la n.306. Espressioni con potenze

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$\small \left\{\left[\left(\dfrac{2}{9}\right)^4·\left(\dfrac{3}{14}\right)^4\right]^2 : \left[\left(-\dfrac{1}{7}\right)^2\right]^4 \right\}·\left[\left(-\dfrac{5}{6}\right)^3 : \left(\dfrac{5}{18}\right)^3\right]^3=$

$\small =\left\{\left[\left(\dfrac{\cancel2^1}{\cancel9_3}\right)^4·\left(\dfrac{\cancel3^1}{\cancel{14}_7}\right)^4\right]^2 : \left(-\dfrac{1}{7}\right)^{2·4} \right\}·\left[\left(-\dfrac{\cancel5^1}{\cancel6_1}\right)^3·\left(\dfrac{\cancel{18}^3}{\cancel5_1}\right)^3\right]^3=$

$\small =\left\{\left[\left(\dfrac{1}{3}\right)^4·\left(\dfrac{1}{7}\right)^4\right]^2 : \left(-\dfrac{1}{7}\right)^8 \right\}·\left[\left(-\dfrac{1}{1}\right)^3·\left(\dfrac{3}{1}\right)^3\right]^3=$

$\small =\left\{\left(\dfrac{1}{3}\right)^{4·2}·\left(\dfrac{1}{7}\right)^{4·2} : \left(-\dfrac{1}{7}\right)^8 \right\}·\left[\left(-\dfrac{3}{1}\right)^3\right]^3=$

$\small =\left\{\left(\dfrac{1}{3}\right)^8·\left(\dfrac{1}{7}\right)^8 : \left(-\dfrac{1}{7}\right)^8 \right\}·\left(-3\right)^{3·3}=$

$\small =\left\{\left(\dfrac{1}{3}\right)^8·\left(-\dfrac{1}{7}\right)^{8-8} \right\}·\left(-3\right)^9=$

$\small =\left\{\left(\dfrac{1}{3}\right)^8·\left(-\dfrac{1}{7}\right)^0 \right\}·\left(-3\right)^9=$

$\small =\left\{\left(\dfrac{1}{3}\right)^8·1 \right\}·\left(-3\right)^9=$

$\small =\left(\dfrac{1}{3}\right)^8·\left(-3\right)^9=$

$\small =3^{-8}·\left(-3\right)^9=$

$\small =-3^{-8+9}=$

$\small =-3^1= -3$

P.s.: Se ti può servire, come esempio, ti metto un altro finale cioè lavorando in croce con gli esponenti:

$\small =\left(\dfrac{1}{3}\right)^8·\left(-\dfrac{3}{1}\right)^9=$

$\small = \dfrac{1^8}{3^8}·-\dfrac{3^9}{1^9}=$

$\small = -\dfrac{3^{9-8}}{1^{8-9}}=$

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

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

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

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

$\small = -3$

nel brogliaccio avevo fatto così ma poi mi sembrava più rapido lo svolgimento, qui ho lasciato tutti gli uno per capire meglio ma tu puoi semplificarli come vuoi.

((2/9)^4·(3/14)^4)^2/((- 1/7)^2)^4·((- 5/6)^3/(5/18)^3)^3=

=(2^4/3^8·(3^4/(2^4·7^4)))^2/(1/7^8)·((- 5^3/(2^3·3^3))/(5^3/(2^3·3^6)))^3=

=(1/(3^4·7^4))^2/(1/7^8)·(- 3^3)^3=

=1/(3^8·7^8)/(1/7^8)·(- 3^9)=

=1/3^8·(- 3^9)= -3