Bài 2 trang 13

Đề bài

Viết các biểu thức sau dưới dạng một luỹ thừa $\left(a>0\right)$:

a) $3.\sqrt{3}.\sqrt[4]{43}.\sqrt[8]{83}$;

b) $\sqrt{a\sqrt{a\sqrt{a}}}$;

c) $\frac{\sqrt{a}.\sqrt[3]{3a}.\sqrt[4]{4a}}{{\left(\sqrt[5]{5a}\right)}^3.a^{\frac{2}{5}}}$.

Lời giải chi tiết

a) $3.\sqrt{3}.\sqrt[4]{43}.\sqrt[8]{83}={3.3}^{\frac{1}{2}}{.3}^{\frac{1}{4}}{.3}^{\frac{1}{8}}=3^{1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}}=3^{\frac{15}{8}}$

b) $\sqrt{a\sqrt{a\sqrt{a}}}=\sqrt{a\sqrt{a.a^{\frac{1}{2}}}}=\sqrt{a\sqrt{a^{1+\frac{1}{2}}}}=\sqrt{a\sqrt{a^{\frac{3}{2}}}}=\sqrt{a.a^{\frac{3}{4}}}=\sqrt{a^{1+\frac{3}{4}}}=\sqrt{a^{\frac{7}{4}}}=a^{\frac{7}{8}}$

c) $\frac{\sqrt{a}.\sqrt[3]{3a}.\sqrt[4]{4a}}{{\left(\sqrt[5]{5a}\right)}^3.a^{\frac{2}{5}}}=\frac{a^{\frac{1}{2}}.a^{\frac{1}{3}}.a^{\frac{1}{4}}}{\sqrt[5]{5a^3}.a^{\frac{2}{5}}}=\frac{a^{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}}}{a^{\frac{3}{5}}.a^{\frac{2}{5}}}=\frac{a^{\frac{13}{12}}}{a^{\frac{3}{5}+\frac{2}{5}}}=\frac{a^{\frac{13}{12}}}{a}=a^{\frac{13}{12}-1}=a^{\frac{1}{12}}$