g) $\frac{3^{4}}{{(3^{- 2})}^{- 3}} = \frac{3^{4}}{{3^{(- 2)}}^{. (- 3)}} = \frac{3^{4}}{3^{6}}$ $= 3^{4 - 6} = 3^{- 2} = \frac{1}{3^{2}} = \frac{1}{9}$

Câu 2/13

Tính giá trị của các biểu thức sau:

a) $\sqrt[3]{0, 001}$ ;

b) $\sqrt[5]{- 32}$ ;

c) $\sqrt[4]{\frac{81}{16}}$ ;

d) $- \sqrt[6]{100^{3}}$ ;

e) $\sqrt[4]{{(\sqrt{3} - 2)}^{4}}$ ;

g) $\sqrt[5]{{(2 - \sqrt{5})}^{5}}$ .

Xem đáp án

Lời giải

a) $\sqrt[3]{0, 001} = \sqrt[3]{{(0, 1)}^{3}} = 0, 1$ .

b) $\sqrt[5]{- 32} = \sqrt[5]{(- 1) .32} = \sqrt[5]{{(- 1)}^{5} {.2}^{5}}$

$= \sqrt[5]{{(- 1)}^{5}} . \sqrt[5]{2^{5}} = (- 1) . 2 = - 2$

c) $\sqrt[4]{\frac{81}{16}} = \sqrt[4]{\frac{3^{4}}{2^{4}}} = \sqrt[4]{{(\frac{3}{2})}^{4}} = \frac{3}{2}$

d) $- \sqrt[6]{100^{3}} = - \sqrt[6]{{(10^{2})}^{3}} = - \sqrt[6]{10^{6}} = - 10$

e) $\sqrt[4]{{(\sqrt{3} - 2)}^{4}} = |\sqrt{3} - 2| = 2 - \sqrt{3}$

g) $\sqrt[5]{{(2 - \sqrt{5})}^{5}} = 2 - \sqrt{5}$

Câu 3/13

Tính giá trị của các biểu thức sau:

a) $\sqrt[4]{125} . \sqrt[4]{5}$ ;

b) $\frac{\sqrt[4]{243}}{\sqrt[4]{3}}$ ;

c) $\frac{\sqrt[3]{3}}{\sqrt[3]{24}}$ ;

d) $\sqrt{\sqrt[3]{64}}$ ;

e) $\sqrt[4]{3 \sqrt[3]{3}}$ ;

g) ${(- \sqrt[6]{4})}^{3}$ .

Xem đáp án

Lời giải

a) $\sqrt[4]{125} . \sqrt[4]{5} = \sqrt[4]{125.5} = \sqrt[4]{5^{3} .5} = \sqrt[4]{5^{4}} = 5$ .

b) $\frac{\sqrt[4]{243}}{\sqrt[4]{3}} = \sqrt[4]{\frac{243}{3}} = \sqrt[4]{81} = \sqrt[4]{3^{4}} = 3$ .

c) $\frac{\sqrt[3]{3}}{\sqrt[3]{24}} = \sqrt[3]{\frac{3}{24}} = \sqrt[3]{\frac{1}{8}} = \sqrt[3]{{(\frac{1}{2})}^{3}} = \frac{1}{2}$ .

d) $\sqrt{\sqrt[3]{64}} = \sqrt[6]{64} = \sqrt[6]{2^{6}} = 2$ .

e) $\sqrt[4]{3 \sqrt[3]{3}} = \sqrt[4]{{(\sqrt[3]{3})}^{3} . \sqrt[3]{3}} = \sqrt[4]{{(\sqrt[3]{3})}^{4}} = \sqrt[3]{3}$ .

g) ${(- \sqrt[6]{4})}^{3} = {(- 1)}^{3} . {(\sqrt[6]{4})}^{3} = - \sqrt[6]{4^{3}}$

$= - \sqrt[6]{{(2^{2})}^{3}} = - \sqrt[6]{2^{6}} = - 2$

Câu 4/13

Tính giá trị của các biểu thức sau:

a) $\sqrt[3]{135} - 5 \sqrt[3]{5}$ ;

b) $\sqrt[4]{\sqrt[3]{81}} + 3 \sqrt[3]{3}$ ;

c) $\sqrt[4]{\sqrt[5]{16}} + \sqrt[5]{64} + 2 \sqrt[5]{2}$ ;

d) ${(\sqrt[4]{5})}^{5} - \sqrt{\sqrt[4]{25}}$ .

Xem đáp án

Lời giải

a) $\sqrt[3]{135} - 5 \sqrt[3]{5} = \sqrt[3]{3^{3} . 5} - 5 \sqrt[3]{5} = \sqrt[3]{3^{3}} . \sqrt[3]{5} - 5 \sqrt[3]{5}$

$= 3 \sqrt[3]{5} - 5 \sqrt[3]{5} = - 2 \sqrt[3]{5}$

b) $\sqrt[4]{\sqrt[3]{81}} + 3 \sqrt[3]{3} = \sqrt[4]{\sqrt[3]{3^{4}}} + 3 \sqrt[3]{3} = \sqrt[3]{3} + 3 \sqrt[3]{3} = 4 \sqrt[3]{3}$

c) $\sqrt[4]{\sqrt[5]{16}} + \sqrt[5]{64} + 2 \sqrt[5]{2} = \sqrt[4]{\sqrt[5]{2^{4}}} + \sqrt[5]{2^{6}} + 2 \sqrt[5]{2}$

$= \sqrt[5]{2} + 2 \sqrt[5]{2} + 2 \sqrt[5]{2} = 5 \sqrt[5]{2}$

d) ${(\sqrt[4]{5})}^{5} - \sqrt{\sqrt[4]{25}} = {(\sqrt[4]{5})}^{4} . \sqrt[4]{5} - \sqrt{\sqrt[4]{5^{2}}}$

$= 5 \sqrt[4]{5} - \sqrt[4]{5} = 4 \sqrt[4]{5}$

Câu 5/13

Không sử dụng máy tính cầm tay, tính giá trị của các biểu thức sau:

a) $8^{- \frac{2}{3}}$ ;

b) $32^{- \frac{2}{5}}$ ;

c) $81^{1, 25}$ ;

d) $1000^{- \frac{5}{3}}$ ;

e) ${(\frac{16}{81})}^{- \frac{1}{4}}$ ;

g) ${(\frac{8}{27})}^{- \frac{2}{3}}$ .

Xem đáp án

Lời giải

a) $8^{- \frac{2}{3}} = {(2^{3})}^{- \frac{2}{3}} = 2^{- 2} = \frac{1}{2^{2}} = \frac{1}{4}$

b) $32^{- \frac{2}{5}} = {(\frac{1}{32})}^{\frac{2}{5}} = \frac{1}{\sqrt[5]{32^{2}}} = \frac{1}{\sqrt[5]{{(2^{5})}^{2}}} = \frac{1}{2^{2}} = \frac{1}{4}$

c) $81^{1, 25} = 81^{\frac{5}{4}} = \sqrt[4]{81^{5}} = \sqrt[4]{{(3^{4})}^{5}} = 3^{5} = 243$

d) $1000^{- \frac{5}{3}} = {(\frac{1}{1000})}^{\frac{5}{3}} = \frac{1}{\sqrt[3]{1000^{5}}} = \frac{1}{\sqrt[3]{{(10^{3})}^{5}}} = \frac{1}{10^{5}} = 10^{- 5}$

e) ${(\frac{16}{81})}^{- \frac{1}{4}} = {(\frac{81}{16})}^{\frac{1}{4}} = \sqrt[4]{\frac{81}{16}} = \sqrt[4]{\frac{3^{4}}{2^{4}}} = \sqrt[4]{{(\frac{3}{2})}^{4}} = \frac{3}{2}$

g) ${(\frac{8}{27})}^{- \frac{2}{3}} = {(\frac{27}{8})}^{\frac{2}{3}} = \sqrt[3]{{(\frac{27}{8})}^{2}}$

$= \sqrt[3]{{[{(\frac{3}{2})}^{3}]}^{2}} = {(\frac{3}{2})}^{2} = \frac{9}{4}$

Câu 6/13

Viết mỗi biểu thức sau dưới dạng một lũy thừa (a > 0):

a) $\sqrt[4]{2^{- 3}}$ ;

b) $\frac{1}{\sqrt[5]{2^{3}}}$ ;

c) ${(\sqrt[5]{3})}^{4}$ ;

d) $\sqrt{a \sqrt[3]{a}}$ ;

e) $\sqrt[3]{a} . \sqrt[4]{a^{3}} : {(\sqrt[6]{a})}^{5}$ ;

g) $a^{\frac{1}{3}} : a^{- \frac{3}{2}} . a^{- \frac{2}{3}}$ .

Xem đáp án

Lời giải

a) $\sqrt[4]{2^{- 3}} = \sqrt[4]{{(\frac{1}{2})}^{3}} = {(\frac{1}{2})}^{\frac{3}{4}} = 2^{- \frac{3}{4}}$

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

c) ${(\sqrt[5]{3})}^{4} = \sqrt[5]{3^{4}} = 3^{\frac{4}{5}}$

d) $\sqrt{a \sqrt[3]{a}} = \sqrt{{(\sqrt[3]{a})}^{3} . \sqrt[3]{a}} = \sqrt{{(\sqrt[3]{a})}^{4}} = \sqrt{\sqrt[3]{a^{4}}} = \sqrt[3]{a^{2}} = a^{\frac{2}{3}}$

e) $\sqrt[3]{a} . \sqrt[4]{a^{3}} : {(\sqrt[6]{a})}^{5} = a^{\frac{1}{3}} . a^{\frac{3}{4}} : a^{\frac{5}{6}} = a^{\frac{1}{3} + \frac{3}{4} - \frac{5}{6}} = a^{\frac{1}{4}}$

g) $a^{\frac{1}{3}} : a^{- \frac{3}{2}} . a^{- \frac{2}{3}} = a^{\frac{1}{3} - (- \frac{3}{2}) + (- \frac{2}{3})} = a^{\frac{1}{3} + \frac{3}{2} - \frac{2}{3}} = a^{\frac{7}{6}}$

Câu 7/13