Esercizio 6 – Espressione con i numeri razionali

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Esercizio.  (\bigstar\largewhitestar\largewhitestar) Semplificare la seguente espressione

    \[\left[\left(-\dfrac{3}{5}\right)^4\cdot \left(\dfrac{5}{9}\right)\cdot \left(-\dfrac{1}{3}\right)^7 \right]: \left[\left(-\dfrac{1}{3}\right)^5\right]^2\]

 

Soluzione

Procediamo come segue

    \[\begin{aligned} 		\left[\left(-\dfrac{3}{5}\right)^4\cdot \left(\dfrac{5}{9}\right)\cdot \left(-\dfrac{1}{3}\right)^7 \right]: \left[\left(-\dfrac{1}{3}\right)^5\right]^2 & = 	\left[\left(-\dfrac{3}{5} \cdot \dfrac{5}{9}\right)^4 \cdot \left(-\dfrac{1}{3}\right)^7 \right]: \left(-\dfrac{1}{3}\right)^{5 \cdot 2} =\\\\ 		& = \left[\left(-\dfrac{\cancel{3}^{\;1}}{\cancel{5}_{\;1}} \cdot \dfrac{\cancel{5}^{\;1}}{\cancel{9}_{\;3}}\right)^4 \cdot \left(-\dfrac{1}{3}\right)^7 \right]: \left(-\dfrac{1}{3}\right)^{10} = \\\\ 		& = \left[\left(-\dfrac{1}{3}\right)^4 \cdot \left(-\dfrac{1}{3}\right)^7 \right]: \left(-\dfrac{1}{3}\right)^{10} = \\\\ 		& = -\left[\left(\dfrac{1}{3}\right)^4 \cdot \left(\dfrac{1}{3}\right)^7 \right]: \left(-\dfrac{1}{3}\right)^{10} = \\\\ 		& = -\left(\dfrac{1}{3}\right)^{4+7} : \left(-\dfrac{1}{3}\right)^{10} = \\\\ 		& = -\left(\dfrac{1}{3}\right)^{11} : \left(\dfrac{1}{3}\right)^{10} = \\\\ 		& = -\left(\dfrac{1}{3}\right)^{11-10} = -\dfrac{1}{3} \end{aligned}\]


Fonte: Algebra Blu con Statistica – Volume 1