Question

Va rog rezolvați problemele astea

Answer 1

9)

$\frac{2^{n+2}\cdot 3^n}{5^n\cdot 2^{n+1}}=\frac{3^{n+2}\cdot2}{9\cdot5^n}\\\\ 2^{n+2-n-1}\cdot\frac{3^n}{5^n}=\frac{3^{n+2}\cdot2}{3^2\cdot5^n}\\\\ 2^1\cdot\frac{3^n}{5^n}=3^{n+2-2}\frac{2}{5^n}\\\\ 2\cdot\frac{3^n}{5^n}=3^n\cdot\frac{2}{5^n}\text{ (A)}$

10)

$\frac{x}{6^{n+2}\cdot 7^n}=\frac{3^{2n}:9^n}{42^n}\\\\ \frac{x}{6^{n+2}\cdot 7^n}=\frac{3^{2n}:3^{2n}}{6^n\codt7^n}\\\\ x\cdot 6^n\cdot7^n=6^{n+2}\cdot7^n\\\\ x=\frac{6^{n+2}\cdot 7^n}{6^n\cdot7^n}=6^{n+2-n}=6^2=36$

11)

$\frac{\overline{abc}}{\overline{bc}}=\frac{5}{1}\\\\ \frac{10^2\cdot a+10\cdot b+c}{10\cdot b+c}=\frac{5}{1}\\\\ \frac{10^2\cdot a+2^2\cdot 5\cdot b+2\cdot c}{10\cdot b+c}=\frac{10^2\cdot a+2\cdot 10\cdot b+2\cdot c}{10\cdot b+c}=\frac{10^2\cdot a+10\cdot b+10\cdot b+c+c}{10\cdot b+c}=\\\\ =\frac{10^2\cdot a+10b+c+10b+c}{10b+c}=\frac{5+10b+c}{10b+c}=\frac{5+1}{1}=\frac{6}{1}$