Құрамында рационал көрсеткішті дәрежелері бар сандық өрнектерді түрлендіріп, амалдарды орындаңыз.
№ 1 Есептеңіз: ${\left( {\dfrac{1}{4}} \right)^{ – \frac{3}{2}}} + 3 \cdot {0,0081^{ – \frac{1}{4}}} + {\left( {\dfrac{1}{{16}}} \right)^{ – 0,75}}$
Шешуі: $$ = {\left( {{2^{ – 2}}} \right)^{ – \frac{3}{2}}} + 3 \cdot {\left( {{{\left( {0,3} \right)}^4}} \right)^{ – \frac{1}{4}}} + {\left( {{2^{ – 4}}} \right)^{ – \frac{3}{4}}} = $$ $$ = {2^{ – 2 \cdot \left( { – \frac{3}{2}} \right)}} + 3 \cdot {0,3^{4 \cdot \left( { – \frac{1}{4}} \right)}} + {2^{ – 4 \cdot \left( { – \frac{3}{4}} \right)}} = $$ $$ = {2^3} + 3 \cdot {0,3^{ – 1}} + {2^3} = 8 + 3 \cdot \frac{{10}}{3} + 8 = 8 + 10 + 8 = 26.$$ Жауабы: $26$
№ 2 Есептеңіз: ${1000^{ – \frac{2}{3}}} + {\left( {\dfrac{1}{{27}}} \right)^{ – \frac{4}{3}}} – {625^{ – 0,75}}$
Шешуі: $$ = {\left( {{{10}^3}} \right)^{ – \frac{2}{3}}} + {\left( {{3^{ – 3}}} \right)^{ – \frac{4}{3}}} – {\left( {{5^4}} \right)^{ – \frac{3}{4}}} = $$ $$ = {10^{3 \cdot \left( { – \frac{2}{3}} \right)}} + {3^{ – 3 \cdot \left( { – \frac{4}{3}} \right)}} – {5^{4 \cdot \left( { – \frac{3}{4}} \right)}} = $$ $$ = {10^{ – 2}} + {3^4} – {5^{ – 3}} = \frac{1}{{100}} + 81 – \frac{1}{{125}} = $$ $$ = 0,01 + 81 – 0,008 = 80,002$$ Жауабы: $80,002$
№ 3 Есептеңіз: ${81^{0,75}} \cdot {32^{ – 0,4}} – {8^{ – \frac{2}{3}}} \cdot {27^{\frac{1}{3}}} + {256^{0,5}}$
Шешуі: $$ = {\left( {{3^4}} \right)^{\frac{3}{4}}} \cdot {\left( {{2^5}} \right)^{ – \frac{2}{5}}} – {\left( {{2^3}} \right)^{ – \frac{2}{3}}} \cdot {\left( {{3^3}} \right)^{\frac{1}{3}}} + {\left( {{2^8}} \right)^{\frac{1}{2}}} = $$ $$ = {3^{4 \cdot \frac{3}{4}}} \cdot {2^{5 \cdot \left( { – \frac{2}{5}} \right)}} – {2^{3 \cdot \left( { – \frac{2}{3}} \right)}} \cdot {3^{3 \cdot \frac{1}{3}}} + {2^{8 \cdot \frac{1}{2}}} = $$ $$ = {3^3} \cdot {2^{ – 2}} – {2^{ – 2}} \cdot {3^1} + {2^4} = 27 \cdot \frac{1}{4} – \frac{1}{4} \cdot 3 + 16 = $$ $$ = \frac{{27}}{4} – \frac{3}{4} + 16 = 16 + \frac{{24}}{4} = 16 + 6 = 22.$$ Жауабы: $22$
№ 4 Есептеңіз: $\sqrt[3]{{ – 5}} \cdot \sqrt 8 \cdot \sqrt[3]{{25}} \cdot \sqrt {32} + \dfrac{{\sqrt[5]{{ – 729}}}}{{\sqrt[5]{3}}}$
Шешуі: $$ = \sqrt[3]{{ – 5 \cdot 25}} \cdot \sqrt {8 \cdot 32} + \sqrt[5]{{\frac{{ – 729}}{3}}} = \sqrt[3]{{ – {5^3}}} \cdot \sqrt {256} + \sqrt[5]{{ – 243}} = $$ $$ = – 5 \cdot 16 + ( – 3) = – 80 – 3 = – 83$$ Жауабы: $-83$
№ 5 Есептеңіз: $4 \cdot {(0,0025)^{ – 0,5}} \cdot \sqrt[3]{{0,001}}$
Шешуі: $$ = 4 \cdot {\left( {{{(0,05)}^2}} \right)^{ – 0,5}} \cdot \sqrt[3]{{(0,1)^3}} = 4 \cdot {0,05^{ – 1}} \cdot 0,1 = 4 \cdot \frac{{100}}{5} \cdot \frac{1}{{10}} = 8$$ Жауабы: $8$
№ 6 Есептеңіз: $5 \cdot \sqrt {0,0004} \cdot {0,216^{ – \frac{1}{3}}}$
Шешуі: $$ = 5 \cdot 0,02 \cdot \frac{1}{{\sqrt[3]{{{{0,6}^3}}}}} = \frac{{0,1}}{{0,6}} = \frac{1}{6}$$ Жауабы: $\dfrac{1}{6}$
№ 7 Есептеңіз: ${64^{\frac{1}{2}}} \cdot {\left( {3\dfrac{3}{8}} \right)^{ – \frac{2}{3}}} \cdot \sqrt {324} $
Шешуі: $$ = \sqrt {64} \cdot {\left( {\frac{{27}}{8}} \right)^{ – \frac{2}{3}}} \cdot 18 = 8 \cdot {\left( {\frac{2}{3}} \right)^{3 \cdot \frac{2}{3}}} \cdot 18 = 8 \cdot \frac{4}{9} \cdot 18 = 64.$$ Жауабы: $64$
№ 8 Теңдеуді шешіңіз: ${\dfrac{{{{(\sqrt[3]{2})}^6} \cdot {2^{ – 6}}}}{{{{\left( {\frac{1}{2}} \right)}^2} \cdot {8^{ – \frac{2}{3}}}}} = x \cdot {{\left( {\dfrac{1}{{\sqrt[3]{2}}}} \right)}^9}}$
Шешуі: $${\frac{{{2^{\frac{6}{3}}} \cdot {2^{ – 6}}}}{{{2^{ – 2}} \cdot {{\left( {{2^3}} \right)}^{ – \frac{2}{3}}}}} = x \cdot {2^{ – \frac{9}{3}}}}$$ $${\frac{{{2^{2 + ( – 6)}}}}{{{2^{ – 2 + ( – 2)}}}} = x \cdot {2^{ – 3}}}$$ $${\frac{{{2^{ – 4}}}}{{{2^{ – 4}}}} = \frac{x}{8}}$$ $$\dfrac{x}{8}=1$$ $$x=8$$ Жауабы: ${x = 8}$
№ 9 Есептеңіз: $\dfrac{{{4^{ – 0,5}} + {{(\sqrt 8 )}^{\frac{2}{3}}} + 2\frac{1}{3}:1\frac{5}{9}}}{{{{\left( {4,8 \cdot 6\frac{2}{3} – 31,75} \right)}^{ – 0,5}}}}$
Шешуі: $$ = \frac{{{{\left( {{2^2}} \right)}^{ – 0,5}} + {{\left( {{2^{\frac{3}{2}}}} \right)}^{\frac{2}{3}}} + \frac{7}{3} \cdot \frac{9}{{14}}}}{{{{\left( {\frac{{24}}{5} \cdot \frac{{20}}{3} – 31,75} \right)}^{ – 0,5}}}} = \frac{{{2^{ – 1}} + {2^1} + \frac{3}{2}}}{{{{0,25}^{ – 0,5}}}} = \frac{4}{{\sqrt 4 }} = \frac{4}{2} = 2.$$ Жауабы: $2$
№ 10$\quad 10\%$-і $ \quad {{{32}^{\frac{2}{5}}} \cdot 0,5 – {{\left( {\sqrt {25} } \right)}^0} + {{\left( { – \dfrac{1}{5}} \right)}^{ – 2}} + {{\left( {\dfrac{2}{3}} \right)}^{ – 4}} \cdot {{\left( {\dfrac{2}{3}} \right)}^3}}$ өрнегінің мәніне тең болатын санды табыңыз.
Шешуі: $${{{32}^{\frac{2}{5}}} \cdot 0,5 – {{\left( {\sqrt {25} } \right)}^0} + {{\left( { – \frac{1}{5}} \right)}^{ – 2}} + {{\left( {\frac{2}{3}} \right)}^{ – 4}} \cdot {{\left( {\frac{2}{3}} \right)}^3} = }$$ $${ = {{\left( {{2^5}} \right)}^{\frac{2}{5}}} \cdot 0,5 – 1 + 25 + {{\left( {\frac{2}{3}} \right)}^{ – 4 + 3}} = }$$ $${ = {2^2} \cdot 0,5 + 24 + \frac{3}{2} = 2 + 24 + 1,5 = 27,5}$$ $${10\% – 27,5}$$ $${100\% – x}$$ $${\frac{{10}}{{100}} = \frac{{27,5}}{x}}$$ $${x = \frac{{100 \cdot 27,5}}{{10}}}$$ Жауабы: $275$
№ 11 Есептеңіз: ${3^{0,3}}:\dfrac{{{3^{ – 0,2}}}}{{1 – {3^{0,5}}}} + \dfrac{2}{{1 – \sqrt 3 }}$
Шешуі: $$ = {3^{0,3}} \cdot \frac{{1 – \sqrt 3 }}{{{3^{ – 0,2}}}} + \frac{{2(1 + \sqrt 3 )}}{{(1 – \sqrt 3 )(1 + \sqrt 3 )}} = $$ $$ = {3^{0,3 – ( – 0,2)}} \cdot (1 – \sqrt 3 ) + \frac{{2(1 + \sqrt 3 )}}{{ – 2}} = $$ $$ = {3^{0,5}}(1 – \sqrt 3 ) – (1 + \sqrt 3 ) = $$ $$ = \sqrt 3 – 3 – 1 – \sqrt 3 = – 4$$ Жауабы: $-4$
№ 12 Есептеңіз: ${16^{ – 0,75}} \cdot {25^{0,5}} + {64^{ – \frac{1}{2}}} \cdot {9^{1,5}} – {\left( {\dfrac{1}{{100}}} \right)^{ – 0,5}}$
Шешуі: $$ = {\left( {{2^4}} \right)^{ – \frac{3}{4}}} \cdot {\left( {{5^2}} \right)^{0,5}} + \frac{1}{{\sqrt {64} }} \cdot {\left( {{3^2}} \right)^{1,5}} – {\left( {{{10}^{ – 2}}} \right)^{ – 0,5}} = $$ $$ = {2^{4 \cdot \left( { – \frac{3}{4}} \right)}} \cdot {5^{2 \cdot 0,5}} + \frac{1}{8} \cdot {3^{2 \cdot 1,5}} – {10^{ – 2 \cdot ( – 0,5)}} = $$ $$ = {2^{ – 3}} \cdot 5 + \frac{{27}}{8} – {10^1} = \frac{5}{8} + \frac{{27}}{8} – 10 = – 6.$$ Жауабы: $-6$
№ 13 Есептеңіз: $\sqrt {2\sqrt {2\sqrt 2 } } $
Шешуі: $$ = \sqrt {\sqrt {{2^2} \cdot 2\sqrt 2 } } = \sqrt {\sqrt {{2^3} \cdot \sqrt 2 } } = \sqrt {\sqrt {\sqrt {{2^6} \cdot 2} } } = \sqrt[{2 \cdot 2 \cdot 2}]{{{2^7}}} = \sqrt[8]{{{2^7}}}$$ Жауабы: $\sqrt[8]{2^7}$
№ 14 Есептеңіз: ${0,027^{ – \frac{1}{3}}} – {\left( { – \dfrac{1}{6}} \right)^{ – 2}} + {256^{0,75}} – {3^{ – 1}} + {10^0}$
Шешуі: $$ = {\left( {{{(0,3)}^3}} \right)^{ – \frac{1}{3}}} – {\left( {{{( – 6)}^{ – 1}}} \right)^{ – 2}} + {256^{\frac{3}{4}}} – \frac{1}{3} + 1 = $$ $$ = {0,3^{3 \cdot \left( { – \frac{1}{3}} \right)}} – {( – 6)^{ – 1 \cdot ( – 2)}} + {\left( {{4^4}} \right)^{\frac{3}{4}}} + \frac{2}{3} = $$ $$ = {0,3^{ – 1}} – {( – 6)^2} + {4^{4 \cdot \frac{3}{4}}} + \frac{2}{3} = \frac{{10}}{3} – 36 + 64 + \frac{2}{3} =$$ $$ = 4 – 36 + 64 = 32$$ Жауабы: $32$
№ 15 Есептеңіз: $2\sqrt[3]{{54}} + \sqrt[3]{{16}} – \sqrt[3]{{250}} + 2\sqrt[3]{{27}} – \sqrt[3]{{128}}$
Шешуі: $$ = 2\sqrt[3]{{27 \cdot 2}} + \sqrt[3]{{8 \cdot 2}} – \sqrt[3]{{125 \cdot 2}} + 2\sqrt[3]{{27}} – \sqrt[3]{{64 \cdot 2}} = $$ $$ = 2 \cdot \sqrt[3]{{27}} \cdot \sqrt[3]{2} + \sqrt[3]{8} \cdot \sqrt[3]{2} – \sqrt[3]{{125}} \cdot \sqrt[3]{2} + 2 \cdot 3 – \sqrt[3]{{64}} \cdot \sqrt[3]{2} = $$ $$ = 2 \cdot 3 \cdot \sqrt[3]{2} + 2\sqrt[3]{2} – 5\sqrt[3]{2} + 6 – 4 \cdot \sqrt[3]{2} = $$ $$ = \sqrt[3]{2}(6 + 2 – 5 – 4) + 6 = 6 – \sqrt[3]{2}$$ Жауабы: $6 – \sqrt[3]{2}$
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