cứu mình với so sánh phân số 11 mũ 70 + 3/ 11 mũ 68+3 và 11 mũ 38 + 3/ 11 mũ 36 + 3

Đáp án:

$\dfrac{11^{70}+3}{11^{68}+3}>\dfrac{11^{38}+3}{11^{36}+3}.$

Giải thích các bước giải:

$\circledast \dfrac{11^{70}+3}{11^{68}+3}\\ =\dfrac{11^{70}+11^2.3-11^2.3+3}{11^{68}+3}\\ =\dfrac{(11^{70}+11^2.3)+(-11^2.3+3)}{11^{68}+3}\\ =\dfrac{11^2(11^{68}+3)-360}{11^{68}+3}\\ =\dfrac{11^2(11^{68}+3)}{11^{68}+3}-\dfrac{360}{11^{68}+3}\\ =11^2-\dfrac{360}{11^{68}+3}\\ \circledast \dfrac{11^{38}+3}{11^{36}+3}\\ =\dfrac{11^{38}+11^2.3-11^2.3+3}{11^{36}+3}\\ =\dfrac{(11^{38}+11^2.3)+(-11^2.3+3)}{11^{36}+3}\\ =\dfrac{11^2(11^{36}+3)-360}{11^{36}+3}\\ =\dfrac{11^2(11^{36}+3)}{11^{36}+3}-\dfrac{360}{11^{36}+3}\\ =11^2-\dfrac{360}{11^{36}+3}\\ 11^{68}>11^{36}\\ \Rightarrow 11^{68}+3>11^{36}+3\\ \Rightarrow \dfrac{360}{11^{68}+3}<\dfrac{360}{11^{36}+3}\\ \Rightarrow -\dfrac{360}{11^{68}+3}>-\dfrac{360}{11^{36}+3}\\ \Rightarrow 11^2-\dfrac{360}{11^{68}+3}>11^2-\dfrac{360}{11^{36}+3}\\ \Leftrightarrow \dfrac{11^{70}+3}{11^{68}+3}>\dfrac{11^{38}+3}{11^{36}+3}.$