Esercizio 9 – Espressione con i numeri naturali

Insieme numerico N

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Esercizio.  (\bigstar\bigstar\largewhitestar)

Semplificare la seguente espressione con i numeri naturali

    \[\left\{ \left[ (15^5:5^5)^2:3^8\right]^5:3^7 - 5^2 \right\}^{11} : \left[(2^4\cdot2^6)^3 : (2^{22}\cdot 2^0)\right]\]

 

Soluzione.

Facciamo i calcoli

    \[\begin{aligned} & \left\{ \left[ ((15:5)^5)^2:3^8\right]^5:3^7 - 5^2 \right\}^{11} : \left[(2^{4+6})^3 : 2^{22+0}\right] = \\ & = \left\{ \left[ (3^5)^2:3^8\right]^5:3^7 - 5^2 \right\}^{11} : \left[(2^{10})^3 : 2^{22}\right] = \\ & = \left\{ \left[ (3^5)^2:3^8\right]^5:3^7 - 5^2 \right\}^{11} : \left[(2^{10})^3 : 2^{22}\right] = \\ & = \left\{ \left[ 3^{5 \cdot 2}:3^8\right]^5:3^7 - 5^2 \right\}^{11} : \left[2^{10 \cdot 3} : 2^{22}\right] = \\ & = \left\{ \left[ 3^{10}:3^8\right]^5:3^7 - 5^2 \right\}^{11} : \left[2^{30} : 2^{22}\right] = \\ & = \left\{ \left[ 3^{10-8}\right]^5:3^7 - 5^2 \right\}^{11} : 2^{30-22} = \\ & = \left\{ \left[ 3^2\right]^5:3^7 - 5^2 \right\}^{11} : 2^{8} = \\  & = \left\{ 3^{2\cdot 5}:3^7 - 5^2 \right\}^{11} : 2^{8} = \\  & = \left\{ 3^{10}:3^7 - 5^2 \right\}^{11} : 2^{8} = \\ & = \left\{ 3^{10-7} - 5^2 \right\}^{11} : 2^{8} = \\ & = \left\{ 3^{3} - 5^2 \right\}^{11} : 2^{8} = \\ & = \left\{ 27-25\right\}^{11} : 2^{8} = \\ & = 2^{11} : 2^{8} = \\ & = 2^{11-8} = \\ & = 2^3 = \\ & = 8 \end{aligned}\]

 

Fonte: L.Sasso – La Matematica a colori (edizione verde)