Bài 2.3 trang 100 SBT giải tích 12

Đề bài

Cho a và b là các số dương. Đơn giản các biểu thức sau:

a) \( \dfrac{a^{\dfrac{4}{3}}\Big( a^{\dfrac{-1}{3}} + a^{\dfrac{2}{3}} \Big)} {a^{\dfrac{1}4}{\Big( a^{\dfrac{3}{4}} + a^{\dfrac{-1}{4}} \Big)}}\)

\( b) \dfrac{ a^{\dfrac{1}{3}}\sqrt{b} + b^{\dfrac{1}{3}}\sqrt{a}}{\sqrt[6]{a} + \sqrt[6]{b}} \)

c) \( \Big( \sqrt[3]{a} + \sqrt[3]{b} \Big)( a^{\dfrac{2}{3}} + b^{\dfrac{2}{3} }- \sqrt[3]{ab} \Big) \)

d)\(\Big( a^{\dfrac{1}{3}} + b^{\dfrac{1}{3}} \Big) : \Big( 2 + \sqrt[3]{\dfrac{a}{b}} + \sqrt[3]{\dfrac{b}{a}}\Big).\)

Lời giải chi tiết

Với a và b là các số dương ta có:

a) \( \dfrac{a^{\dfrac{4}{3}}\Big( a^{\dfrac{-1}{3}} + a^{\dfrac{2}{3}} \Big)} {a^{\dfrac{1}{4}}\Big( a^{\dfrac{3}{4}} + a^{\dfrac{-1}{4}} \Big)}\)

\(= \dfrac{a^{\dfrac{4}{3}}. a^{\dfrac{-1}{3}} + a^{\dfrac{2}{3}}.a^{\dfrac{4}{3}} } {a^{\dfrac{1}{4}}. a^{\dfrac{3}{4}} + a^{\dfrac{1}{4}}. a^{\dfrac{-1}{4}}}\)

\(= \dfrac{a^1 + a^2}{a^1 + a^0} = \dfrac{a\Big( a + 1\Big)}{a + 1} =a \)

b) \( \dfrac{ a^{\dfrac{1}{3}}\sqrt{b} + b^{\dfrac{1}{3}}\sqrt{a}}{\sqrt[6]{a} + \sqrt[6]{b}} \)

\(= \dfrac{a^{\dfrac{1}{3}}b^{\dfrac{1}{2}} + b^{\dfrac{1}{3}}a^{\dfrac{1}{2}}}{ a^{\dfrac{1}{6}} + b^{\dfrac{1}{6}}}\)

\(= \dfrac{a^{\dfrac{1}{3}}b^{\dfrac{1}{3}}\Big(b^{\dfrac{1}{2} - \dfrac{1}{3}}+ a^{\dfrac{1}{2} - \dfrac{1}{3}} \Big)}{a^{\dfrac{1}{6}} + b^{\dfrac{1}{6}}}\)

\(= \dfrac{a^{\dfrac{1}{3}}b^{\dfrac{1}{3}}\Big(b^{\dfrac{1}{6}} +a^{ \dfrac{1}{6}} \Big)}{a^{\dfrac{1}{6}} + b^{\dfrac{1}{6}}}\)

\(=\sqrt[3]{ab} \)

c) \(\Big( \sqrt[3]{a} + \sqrt[3]{b} \Big)( a^{\dfrac{2}{3}} + b^{\dfrac{2}{3} }- \sqrt[3]{ab} \Big) \)

\(= \Big( a^{\dfrac{1}{3}} + b^{\dfrac{1}{3}}\Big) \Big( a^{\dfrac{2}{3}} - a^{\dfrac{1}{3}}. b^{\dfrac{1}{3}}+ b^{\dfrac{2}{3}}\Big)\)

\(= {\Big( a^{\dfrac{1}{3}} \Big) }^{3} + {\Big( b^{\dfrac{1}{3}} \Big) }^{3}\)

\(= a + b\)

d)\( \Big( a^{\dfrac{1}{3}} + b^{\dfrac{1}{3}} \Big) : \Big( 2 + \sqrt[3]{\dfrac{a}{b}} + \sqrt[3]{\dfrac{b}{a}}\Big) \)

\(=\dfrac{ a^{\dfrac{1}{3}} + b^{\dfrac{1}{3}} }{ \dfrac{{2\sqrt[3]{ab} + \sqrt[3]{a^2} + \sqrt[3]{b^2}}}{\sqrt[3]{ab}}} \)

\(= \dfrac{\Big(\sqrt[3]{a} +\sqrt[3]{b}\Big)\sqrt[3]{ab}}{\Big(\sqrt[3]{a} +\sqrt[3]{b}\Big)^{2} } \)

\(= \dfrac{\sqrt[3]{ab}}{\sqrt[3]{a} +\sqrt[3]{b}}\)