Cho a, b là những số thực dương. Rút gọn các biểu thức

Giải Toán 12 Bài 1: Lũy thừa

Bài 4 trang 56 Toán lớp 12 Giải tích:

a) $\frac{a^{\frac{4}{3}}\left(a^{-\frac{1}{3}}+a^{\frac{2}{3}}\right)}{a^{\frac{1}{4}}\left(a^{\frac{3}{4}}+a^{-\frac{1}{4}}\right)}$;

b) $\frac{b^{\frac{1}{5}}\left(\sqrt[5]{b^4}-\sqrt[5]{b^{-1}}\right)}{b^{\frac{2}{3}}\left(\sqrt[3]{b}-\sqrt[3]{b^{-2}}\right)}$;

c) $\frac{a^{\frac{1}{3}}{b}^{-\frac{1}{3}}-a^{-\frac{1}{3}}{b}^{\frac{1}{3}}}{\sqrt[3]{a^2}-\sqrt[3]{b^2}}$;

d) $\frac{a^{\frac{1}{3}}\sqrt{b}+b^{\frac{1}{3}}\sqrt{a}}{\sqrt[6]{a}+\sqrt[6]{b}}$.

Lời giải:

a)

 $\frac{a^{\frac{4}{3}}\left(a^{-\frac{1}{3}}+a^{\frac{2}{3}}\right)}{a^{\frac{1}{4}}\left(a^{\frac{3}{4}}+a^{-\frac{1}{4}}\right)}=\frac{a^{\frac{4}{3}}.a^{\frac{-1}{3}}+a^{\frac{4}{3}}.a^{\frac{2}{3}}}{a^{\frac{1}{4}}.a^{\frac{3}{4}}+a^{\frac{1}{4}}.a^{\frac{-1}{4}}}=\frac{a^{\frac{4}{3}-\frac{1}{3}}+a^{\frac{4}{3}+\frac{2}{3}}}{a^{\frac{1}{4}+\frac{3}{4}}+a^{\frac{1}{4}-\frac{1}{4}}}=\frac{a+a^2}{a+1}=\frac{a\left(a+1\right)}{a+1}=a$

b) $\frac{b^{\frac{1}{5}}\left(\sqrt[5]{b^4}-\sqrt[5]{b^{-1}}\right)}{b^{\frac{2}{3}}\left(\sqrt[3]{b}-\sqrt[3]{b^{-2}}\right)}$

$=\frac{b^{\frac{1}{5}}\left(b^{\frac{4}{5}}-b^{\frac{-1}{5}}\right)}{b^{\frac{2}{3}}\left(b^{\frac{1}{3}}-b^{-\frac{2}{3}}\right)}=\frac{b^{\frac{1}{5}}.b^{\frac{4}{5}}-b^{\frac{1}{5}}.b^{\frac{-1}{5}}}{b^{\frac{2}{3}}.b^{\frac{1}{3}}-b^{\frac{2}{3}}.b^{\frac{-2}{3}}}=\frac{b^{\frac{1}{5}+\frac{4}{5}}-b^{\frac{1}{5}-\frac{1}{5}}}{b^{\frac{2}{3}+\frac{1}{3}}-b^{\frac{2}{3}-\frac{2}{3}}}=\frac{b-1}{b-1}=1$

c) $\frac{a^{\frac{1}{3}}{b}^{-\frac{1}{3}}-a^{-\frac{1}{3}}{b}^{\frac{1}{3}}}{\sqrt[3]{a^2}-\sqrt[3]{b^2}}$

$=\frac{a^{-\frac{1}{3}}{b}^{-\frac{1}{3}}\left(a^{\frac{2}{3}}-b^{\frac{2}{3}}\right)}{a^{\frac{2}{3}}-b^{\frac{2}{3}}}=a^{-\frac{1}{3}}.b^{-\frac{1}{3}}={\left(a.b\right)}^{-\frac{1}{3}}=\frac{1}{{\left(ab\right)}^{\frac{1}{3}}}=\frac{1}{\sqrt[3]{ab}}$

d) $\frac{a^{\frac{1}{3}}\sqrt{b}+b^{\frac{1}{3}}\sqrt{a}}{\sqrt[6]{a}+\sqrt[6]{b}}$

$=\frac{a^{\frac{1}{3}}.b^{\frac{1}{2}}+b^{\frac{1}{3}}.a^{\frac{1}{2}}}{a^{\frac{1}{6}}+b^{\frac{1}{6}}}=\frac{a^{\frac{1}{3}}{b}^{\frac{1}{3}}\left(b^{\frac{1}{6}}+a^{\frac{1}{6}}\right)}{a^{\frac{1}{6}}+b^{\frac{1}{6}}}=a^{\frac{1}{3}}{b}^{\frac{1}{3}}={\left(ab\right)}^{\frac{1}{3}}=\sqrt[3]{ab}$