Bài 1: Lũy thừa
Bài 2.3 (Sách bài tập trang 95)
Cho a và b là các số dương. Đơn giản các biểu thức sau :
a) \(\dfrac{a^{\dfrac{4}{3}}\left(a^{-\dfrac{1}{3}}+a^{\dfrac{2}{3}}\right)}{a^{\dfrac{1}{4}}\left(a^{\dfrac{3}{4}}+a^{-\dfrac{1}{4}}\right)}\)
b) \(\dfrac{a^{\dfrac{1}{3}}\sqrt{b}+b^{\dfrac{1}{3}}\sqrt{a}}{\sqrt[6]{a}+\sqrt[6]{b}}\)
c) \(\left(\sqrt[3]{a}+\sqrt[3]{b}\right)\left(a^{\dfrac{2}{3}}+b^{\dfrac{2}{3}}-\sqrt[3]{ab}\right)\)
d) \(\left(a^{\dfrac{1}{3}}+b^{\dfrac{1}{3}}\right):\left(2+\sqrt[3]{\dfrac{a}{b}}+\sqrt[3]{\dfrac{b}{a}}\right)\)
Hướng dẫn giải
a)
\(A=\dfrac{a^{\dfrac{4}{3}}\left(a^{-\dfrac{1}{3}}+a^{\dfrac{2}{3}}\right)}{a^{\dfrac{1}{4}}\left(a^{\dfrac{3}{4}}+a^{-\dfrac{1}{4}}\right)}=\dfrac{a^{\left(\dfrac{4}{3}-\dfrac{1}{3}\right)+}a^{\left(\dfrac{4}{3}+\dfrac{2}{3}\right)}}{a^{\left(\dfrac{1}{4}+\dfrac{3}{4}\right)}+a^{\left(\dfrac{1}{4}-\dfrac{1}{4}\right)}}=\dfrac{a+a^2}{a+1}=\dfrac{a\left(a+1\right)}{a+1}\)
\(a>0\Rightarrow a+1\ne0\) \(\Rightarrow A=a\)
Bài 2.5 (Sách bài tập trang 96)
Hãy so sánh các cặp số sau :
a) \(\sqrt{17}\) và \(\sqrt[3]{28}\)
b) \(\sqrt[4]{13}\) và \(\sqrt[5]{23}\)
c) \(\left(\dfrac{1}{3}\right)^{\sqrt{3}}\) và \(\left(\dfrac{1}{3}\right)^{\sqrt{2}}\)
d) \(4^{\sqrt{5}}\) và \(4^{\sqrt{7}}\)
Hướng dẫn giải
a) \(\left(\sqrt{17}\right)^6=\sqrt{\left(17^3\right)^2}=17^3=4913\)
\(\left(\sqrt[3]{28}\right)^6=\sqrt[3]{\left(28^2\right)^3}=28^2=784\)
=> \(\left(\sqrt{17}\right)^6>\left(\sqrt[3]{28}\right)^6\)
=> \(\sqrt{17}>\sqrt[3]{28}\)
Bài 2.4 (Sách bài tập trang 96)
Hãy so sánh mỗi số sau với 1 :
a) \(2^{-2}\)
b) \(\left(0,013\right)^{-1}\)
c) \(\left(\dfrac{2}{7}\right)^5\)
d) \(\left(\dfrac{1}{2}\right)^{\sqrt{3}}\)
e) \(\left(\dfrac{\pi}{4}\right)^{\sqrt{5}-2}\)
g) \(\left(\dfrac{1}{3}\right)^{\sqrt{8}-3}\)
Hướng dẫn giải
a) \(2^{-2}=\dfrac{1}{2^2}< 1\)
b) \(\left(0,013\right)^{-1}=\dfrac{1}{0,013}>1\)
c) \(\left(\dfrac{2}{7}\right)^5=\dfrac{2^5}{7^5}< 1\)
d) \(\left(\dfrac{1}{2}\right)^{\sqrt{3}}=\dfrac{1}{2^{\sqrt{3}}}< \dfrac{1}{2^{\sqrt{1}}}=\dfrac{1}{2}< 1\)
e) vì \(0< \dfrac{\pi}{4}< 1\)
Suy ra \(\left(\dfrac{\pi}{4}\right)^{\sqrt{5}-2}=\dfrac{\left(\dfrac{\pi}{4}\right)^{\sqrt{5}}}{\left(\dfrac{\pi}{2}\right)^2}>\dfrac{\left(\dfrac{\pi}{4}\right)^{\sqrt{4}}}{\left(\dfrac{\pi}{4}\right)^2}=1\)
f) Vì \(0< \dfrac{1}{3}< 1\)
Nên \(\left(\dfrac{1}{3}\right)^{\sqrt{8}-3}>\left(\dfrac{1}{3}\right)^{\sqrt{9}-3}=\left(\dfrac{1}{3}\right)^0=1\)
Bài 2.1 (Sách bài tập trang 95)
Tính :
a) \(2^{2-3\sqrt{5}}.8^{\sqrt{5}}\)
b) \(3^{1+2\sqrt[3]{2}}:9^{\sqrt[3]{2}}\)
c) \(\dfrac{10^{2+\sqrt{7}}}{2^{2+\sqrt{7}}.5^{1+\sqrt{7}}}\)
d) \(\left(4^{2\sqrt{3}}-4^{\sqrt{3}-1}\right).2^{-2\sqrt{3}}\)
Hướng dẫn giải
a)
\(A=2^{2-3\sqrt{5}}.8^{\sqrt{5}}=2^{2-3\sqrt{5}}.2^{3\sqrt{5}}=2^{\left(2-3\sqrt{5}\right)+3\sqrt{5}}=2^2=4\)
\(A=4\)
d)
\(D=\left(4^{2\sqrt{3}}-4^{\sqrt{3}-1}\right).2^{-2\sqrt{3}}=2^{4\sqrt{3}-2\sqrt{3}}-2^{2\sqrt{3}-2-2\sqrt{3}}\)
\(D=2^{2\sqrt{3}}-\dfrac{1}{4}\)
Bài 2.2 (Sách bài tập trang 95)
Tính :
a) \(\left(\dfrac{1}{16}\right)^{-\dfrac{3}{4}}+810000^{0,25}-\left(7\dfrac{19}{32}\right)^{\dfrac{1}{5}}\)
b) \(\left(0,001\right)^{-\dfrac{1}{3}}-2^{-2}.64^{\dfrac{2}{3}}-8^{-1\dfrac{1}{3}}\)
c) \(27^{\dfrac{2}{3}}-\left(-2\right)^{-2}+\left(3\dfrac{3}{8}\right)^{-\dfrac{1}{3}}\)
d) \(\left(-0,5\right)^{-4}-625^{0,25}-\left(2\dfrac{1}{4}\right)^{-1\dfrac{1}{2}}\)
Hướng dẫn giải
b) \(=\left(0,1\right)^{3.\left(-\dfrac{1}{3}\right)}-2^{-2}.2^{6.\dfrac{2}{3}}-\left[\left(2\right)^3\right]^{-\dfrac{4}{3}}\)
\(=0,1^{-1}-2^2-2^{-4}\)
\(=10-4-\dfrac{1}{16}\)
\(=\dfrac{95}{16}\)