Question

a) Tính

A=\(\frac{2^{12}.3^5-4^6.9^2}{\left(2^2.3\right)^6+8^4.3^5}-\frac{5^{10}.7^3-2^5.49^2}{\left(125.7\right)^3+5^9.14^3}\)

b) Chứng tỏ với mọi số nguyên n thì:

\(3^{n-2}-2^{n+2}+3^n-2^n⋮10\)

Answer 1

a:Sửa đề:  \(A=\dfrac{2^{12}\cdot3^5-2^{12}\cdot3^3}{2^{12}\cdot3^6+2^{12}\cdot3^5}-\dfrac{5^{10}\cdot7^3-25^5\cdot7^4}{5^9\cdot7^3+5^9\cdot2^3\cdot7^3}\)

\(=\dfrac{2^{12}\cdot3^3\cdot\left(3^2-1\right)}{2^{12}\cdot3^5\left(3+1\right)}-\dfrac{5^{10}\cdot7^3-5^{10}\cdot7^4}{5^9\cdot7^3\cdot9}\)

\(=\dfrac{1}{9}\cdot\dfrac{8}{4}-\dfrac{5^{10}\cdot7^3\cdot\left(1-7\right)}{5^9\cdot7^3\cdot9}\)

\(=\dfrac{2}{9}-\dfrac{5\cdot\left(-2\right)}{3}=\dfrac{2}{9}+\dfrac{10}{3}=\dfrac{2+30}{9}=\dfrac{32}{9}\)

b: Sửa đề: \(3^{n+2}-2^{n+2}+3^n-2^n\)

\(=3^n\cdot9+3^n-2^n\cdot4-2^n\)

\(=3^n\cdot10-2^n\cdot5\)

\(=10\left(3^n-2^{n-1}\right)⋮10\)