Question

$( {10}^{12} + 25)^{2} - ( {10}^{12} - 25)^{2} = {10}^{n}$
O natural n para o qual... é:​

Answer 1

(10¹² + 25)² - (10¹² - 25)² = 10ⁿ

10²⁴+50.10¹²+625-(10²⁴-50.10¹²+625)=10ⁿ

10²⁴+50.10¹²+625 -10²⁴+50.10¹²-625=10ⁿ

100.10¹² = 10ⁿ

10². 10¹² = 10ⁿ

10²+¹² = 10ⁿ

10¹⁴ = 10ⁿ

n = 14

Answer 2

Resposta:

$\boxed{\mathtt{14}}$

Explicação passo-a-passo:

Lembremo-nos que:

$\displaystyle \boxed{\mathtt{a^2 - b^2 = (a + b) \cdot (a - b)}}$

Isto posto,

$\\ \displaystyle \mathsf{\left ( 10^{12} + 25 \right )^2 - \left ( 10^{12} - 25 \right )^2 = 10^n} \\\\ \mathsf{\left [ \left ( 10^{12} + 25 \right ) + \left ( 10^{12} - 25 \right ) \right ] \cdot \left [ \left ( 10^{12} + 25 \right ) - \left ( 10^{12} - 25 \right ) \right ] = 10^n} \\\\ \mathsf{\left ( 10^{12} + 25 + 10^{12} - 25 \right ) \cdot \left ( 10^{12} + 25 - 10^{12} + 25 \right ) = 10^n} \\\\ \mathsf{2 \cdot 10^{12} \cdot 50 = 10^n} \\\\ \mathsf{2 \cdot 10^{12} \cdot \left ( 5 \cdot 10 \right ) = 10^n} \\\\ \mathsf{2 \cdot 5 \cdot 10^{12} \cdot 10 = 10^n} \\\\ \mathsf{10 \cdot 10^{12} \cdot 10 = 10^n} \\\\ \mathsf{10^{1 + 12 + 1} = 10^n} \\\\ \mathsf{10^{14} = 10^n} \\\\ \boxed{\boxed{\mathsf{n = 14}}}$