Рационал көрсеткішті сандық өрнектерді түрлендіру

Рационал көрсеткіші бар сандық өрнектерді түрлендіру

1.14. Есептеңіз:

1) $\left(\dfrac{1}{\sqrt{3}}\right)^3 \cdot \sqrt[4]{3} \cdot\left(\dfrac{1}{3}\right)^2$

Шешуі

$${\left( {\frac{1}{{\sqrt 3 }}} \right)^3} \cdot \sqrt[4]{3} \cdot {\left( {\frac{1}{3}} \right)^2} = {\left( {{3^{ - \frac{1}{2}}}} \right)^3} \cdot {3^{\frac{1}{4}}} \cdot {\left( {{3^{ - 1}}} \right)^2} = $$ $$ = {3^{ - \frac{3}{2}}} \cdot {3^{\frac{1}{4}}} \cdot {3^{ - 2}} = {3^{ - \frac{{13}}{4}}} = {3^{ - 3,25}}$$

2) $\left(\dfrac{1}{9}\right)^{\frac{11}{10}}:\left(\dfrac{1}{9}\right)^{\frac{6}{5}} \cdot \sqrt[5]{27^3}$

Шешуі

$${\left( {\frac{1}{9}} \right)^{\frac{{11}}{{10}}}}:{\left( {\frac{1}{9}} \right)^{\frac{6}{5}}} \cdot \sqrt[5]{{{{27}^3}}} = {\left( {{3^{ - 2}}} \right)^{\frac{{11}}{{10}}}}:{\left( {{3^{ - 2}}} \right)^{\frac{6}{5}}} \cdot {3^{\frac{9}{5}}} = {3^{ - \frac{{11}}{5} + \frac{{12}}{5} + \frac{9}{5}}} = {3^2} = 9$$

3) $0,0016^{\frac{3}{4}}+0,04^{-\frac{1}{2}}-0,216^{-\frac{2}{3}} \cdot 9$

Шешуі

$${0,0016^{ - \frac{3}{4}}} + {0,04^{ - \frac{1}{2}}} - {0,216^{ - \frac{2}{3}}} \cdot 9 = {\left( {{{0,2}^4}} \right)^{ - \frac{3}{4}}} + {\left( {{{0,2}^2}} \right)^{ - \frac{1}{2}}} - {\left( {{{0,6}^3}} \right)^{ - \frac{2}{3}}} \cdot 9 = $$ $$ = {0,2^{ - 3}} + {0,2^{ - 1}} - {0,6^{ - 2}} \cdot 9 = 125 + 5 - \frac{{25}}{9} \cdot 9 = 105$$

4) $64^{-\frac{5}{6}}-(0,125)^{-\frac{1}{3}}-32 \cdot 2^{-4} \cdot 16^{-1 \frac{1}{2}}+\left(3^0\right)^4 \cdot 4$

Шешуі

$${64^{ - \frac{5}{6}}} - {(0,125)^{ - \frac{1}{3}}} - 32 \cdot {2^{ - 4}} \cdot {16^{ - 1\frac{1}{2}}} + {\left( {{3^0}} \right)^4} \cdot 4 = $$ $$ = {\left( {{2^6}} \right)^{ - \frac{5}{6}}} - {\left( {{2^{ - 3}}} \right)^{ - \frac{1}{3}}} - {2^5} \cdot {2^{ - 4}} \cdot {\left( {{2^4}} \right)^{ - \frac{3}{2}}} + 1 \cdot 4 = $$ $$ = {2^{ - 5}} - 2 - {2^{ - 5}} + 4 = 2$$

5) $81^{0,75} \cdot 32^{-0,4}-8^{-\frac{2}{3}} \cdot 27^{\frac{1}{3}}+256^{0,5}$

Шешуі

$${81^{0,75}} \cdot {32^{ - 0,4}} - {8^{ - \frac{2}{3}}} \cdot {27^{\frac{1}{3}}} + {256^{0,5}} = {\left( {{3^4}} \right)^{0,75}} \cdot {\left( {{2^5}} \right)^{ - 0,4}} - {\left( {{2^3}} \right)^{ - \frac{2}{3}}} \cdot {\left( {{3^3}} \right)^{\frac{1}{3}}} + $$ $$ + \sqrt {256} = {3^3} \cdot {2^{ - 2}} - \left( {{2^{ - 2}}} \right) \cdot 3 + 16 = \frac{{27}}{4} - \frac{3}{4} + 16 = 22$$

6) $1000^{-\frac{2}{3}}+\left(\dfrac{1}{27}\right)^{-\frac{4}{3}}-625^{-0.75}$

Шешуі

$${{{1000}^{ - \frac{2}{3}}} + {{\left( {\frac{1}{{27}}} \right)}^{ - \frac{4}{3}}} - {{625}^{ - 0,75}} = {{\left( {{{10}^3}} \right)}^{ - \frac{2}{3}}} + {{\left( {{3^{ - 3}}} \right)}^{ - \frac{4}{3}}} - {{\left( {{5^4}} \right)}^{ - 0,75}} = }$$ $${ = {{10}^{ - 2}} + {3^4} - {5^{ - 3}} = 0,01 + 81 - 0,008 = 81,002}$$

7) $\left(\sqrt[4]{32 \cdot \sqrt[3]{4}}+\sqrt[4]{64 \cdot \sqrt[3]{\dfrac{1}{2}}}-3 \cdot \sqrt[3]{2 \cdot \sqrt[4]{2}}\right) \cdot \dfrac{3}{\sqrt[12]{2^5}}$

Шешуі

$$\left( {\sqrt[4]{{32 \cdot \sqrt[3]{4}}} + \sqrt[4]{{64 \cdot \sqrt[3]{{\frac{1}{2}}}}} - 3 \cdot \sqrt[3]{{2 \cdot \sqrt[4]{2}}}} \right) \cdot \frac{3}{{\sqrt[{12}]{{{2^5}}}}} = $$ $$ = \left( {{2^{\frac{5}{4}}} \cdot {2^{\frac{2}{{12}}}} + {2^{\frac{6}{4}}} \cdot {2^{ - \frac{1}{{12}}}} - 3 \cdot {2^{\frac{1}{3}}} \cdot {2^{\frac{1}{{12}}}}} \right) \cdot \frac{3}{{{2^{\frac{5}{{12}}}}}} = $$ $${ = \left( {{2^{\frac{{17}}{{12}}}} + {2^{\frac{{17}}{{12}}}} - 3 \cdot {2^{\frac{5}{{12}}}}} \right) \cdot 3 \cdot {2^{ - \frac{5}{{12}}}} = {2^{\frac{5}{{12}}}}(2 + 2 - 3) \cdot {2^{ - \frac{5}{{12}}}} = 3}$$

8) $\dfrac{1}{\sqrt{2}-1}-2^{0,2} \cdot \dfrac{1-2^{0,5}}{2^{-0,3}}$

Шешуі

$$\frac{1}{{\sqrt 2 - 1}} - {2^{0,2}} \cdot \frac{{1 - {2^{0,5}}}}{{{2^{ - 0,3}}}} = \frac{1}{{\sqrt 2 - 1}} - {2^{0,5}}\left( {1 - {2^{0,5}}} \right) = $$ $$ = \frac{{\sqrt 2 + 1}}{{(\sqrt 2 - 1)(\sqrt 2 + 1)}} - {2^{0.5}} + 2 = {2^{0,5}} + 1 - {2^{0,5}} + 2 = 3$$

9) $\left(\dfrac{15 \cdot 5^{\frac{1}{2}}}{125^{-\frac{1}{3}}}-2 \cdot 7^{\frac{1}{2}} \cdot 49^{\frac{1}{4}}\right) \cdot\left(\left(\dfrac{1}{81}\right)^{-\frac{1}{4}}+45^{\frac{1}{2}}\right)-183 \sqrt{5}$

Шешуі

$$\left( {\frac{{15 \cdot {5^{\frac{1}{2}}}}}{{{{125}^{ - \frac{1}{3}}}}} - 2 \cdot {7^{\frac{1}{2}}} \cdot {{49}^{\frac{1}{4}}}} \right) \cdot \left( {{{\left( {\frac{1}{{81}}} \right)}^{ - \frac{1}{4}}} + {{45}^{\frac{1}{2}}}} \right) - 183\sqrt 5 = $$ $$ = \left( {\frac{{3 \cdot {5^{\frac{3}{2}}}}}{{{{\left( {{5^3}} \right)}^{ - \frac{1}{3}}}}} - 2 \cdot {7^{\frac{1}{2}}} \cdot {7^{\frac{1}{2}}}} \right)\left( {{{\left( {{3^{ - 4}}} \right)}^{ - \frac{1}{4}}} + 3\sqrt 5 } \right) - 183\sqrt 5 = $$ $${ = (3 \cdot 5\sqrt 5 \cdot 5 - 2 \cdot 7)(3 + 3\sqrt 5 ) - 183\sqrt 5 = }$$ $$ = (75\sqrt 5 - 14)(3 + 3\sqrt 5 ) - 183\sqrt 5 = $$ $$ = 225\sqrt 5 - 42 + 1125 - 42\sqrt 5 - 183\sqrt 5 = 1083$$

10) $\left(15 \cdot 4^{-2}+\dfrac{2^{-4} \cdot \sqrt{11}}{121^{0,25}}\right) \cdot\left(\dfrac{\left(1+9^{0,25}\right)(\sqrt{3}-1)}{4}\right)^{-1}$

Шешуі

$${\left( {15 \cdot {4^{ - 2}} + \frac{{{2^{ - 4}} \cdot \sqrt {11} }}{{{{121}^{0,25}}}}} \right) \cdot {{\left( {\frac{{\left( {1 + {9^{0,25}}} \right)\left( {\sqrt 3 - 1} \right)}}{4}} \right)}^{ - 1}} = }$$ $$ = \left( {\frac{{15}}{{16}} + \frac{{1 \cdot \sqrt {11} }}{{16 \cdot \sqrt {11} }}} \right)\left( {\frac{4}{{\left( {1 + \sqrt 3 } \right)\left( {\sqrt 3 - 1} \right)}}} \right) = $$ $$ = \left( {\frac{{15}}{{16}} + \frac{1}{{16}}} \right)\left( {\frac{4}{{{{\left( {\sqrt 3 } \right)}^2} - {1^2}}}} \right) = \frac{4}{2} = 2$$



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