Determina o valor das expressões numéricas
Números reais: Matematicamente Falando 8 - Pág. 34 Ex. 8
Enunciado
Aplicando, sempre que possível, as regras da multiplicação e da divisão de potências, determina o valor das expressões numéricas seguintes:
| \[{{2^3} \times {{( – 3)}^3}}\] | \[{{{\left( { – 2} \right)}^2} \times ( – 2) – 3}\] | \[{{{\left( { – 7} \right)}^2} \div {{( – 1)}^2}}\] |
| \[{{{\left( { – 5} \right)}^9} \div {{( – 5)}^{11}}}\] | \[{{{({{10}^3})}^{ – 2}}}\] | \[{{3^2} \times {5^2}}\] |
| \[{{{\left( { – 1} \right)}^5} \times {2^5}}\] | \[{{{\left( { – \frac{1}{3}} \right)}^{ – 4}} \times {3^{ – 4}}}\] | \[{{2^2} \times {{\left( { – 2} \right)}^3}}\] |
| \[{ – \frac{2}{3} \times {{\left( {\frac{3}{2}} \right)}^{ – 3}}}\] | \[{{{\left( { – 4} \right)}^6} \div {2^6}}\] | \[{\left( { – \frac{1}{2}} \right) \div {{\left( { – \frac{1}{3}} \right)}^4}}\] |
| \[{{2^3} \times {{\left( { – 2} \right)}^{ – 4}}}\] | \[{{{\left( { – 3} \right)}^5} \div {3^5}}\] | \[{{{\left( { – 1} \right)}^{102}}}\] |
| \[{{2^3} + {2^4}}\] | \[{{3^{ – 2}} – {{\left( { – 3} \right)}^{ – 2}}}\] | \[{{{\left( { – 2} \right)}^2} + {{\left( { – 3} \right)}^2}}\] |
Resolução
Aplicando, sempre que possível, as regras da multiplicação e da divisão de potências, temos:
| \[\begin{array}{*{20}{l}}{{2^3} \times {{( – 3)}^3}}& = &{{{\left( {2 \times \left( { – 3} \right)} \right)}^3}}\\{}& = &{{{\left( { – 6} \right)}^3}}\\{}& = &{ – 216}\end{array}\] | \[\begin{array}{*{20}{l}}{{{\left( { – 2} \right)}^2} \times ( – 2) – 3}& = &{{{\left( { – 2} \right)}^{2 + 1}} – 3}\\{}& = &{{{\left( { – 2} \right)}^3} – 3}\\{}& = &{ – 8 – 3}\\{}& = &{ – 11}\end{array}\] | \[\begin{array}{*{20}{l}}{{{\left( { – 7} \right)}^2} \div {{( – 1)}^2}}& = &{{{\left( { – 7 \div \left( { – 1} \right)} \right)}^2}}\\{}& = &{{7^2}}\\{}& = &{49}\end{array}\] |
| \[\begin{array}{*{20}{l}}{{{\left( { – 5} \right)}^9} \div {{( – 5)}^{11}}}& = &{{{\left( { – 5} \right)}^{9 – 11}}}\\{}& = &{{{\left( { – 5} \right)}^{ – 2}}}\\{}& = &{{{\left( { – \frac{1}{5}} \right)}^2}}\\{}& = &{\frac{1}{{25}}}\end{array}\] | \[\begin{array}{*{20}{l}}{{{({{10}^3})}^{ – 2}}}& = &{{{10}^{3 \times \left( { – 2} \right)}}}\\{}& = &{{{10}^{ – 6}}}\\{}& = &{{{\left( {\frac{1}{{10}}} \right)}^6}}\\{}& = &{\frac{1}{{1000000}}}\end{array}\] | \[\begin{array}{*{20}{l}}{{3^2} \times {5^2}}& = &{{{\left( {3 \times 5} \right)}^2}}\\{}& = &{{{15}^2}}\\{}& = &{225}\end{array}\] |
| \[\begin{array}{*{20}{l}}{{{\left( { – 1} \right)}^5} \times {2^5}}& = &{{{\left( { – 1 \times 2} \right)}^5}}\\{}& = &{{{\left( { – 2} \right)}^5}}\\{}& = &{ – 32}\end{array}\] | \[\begin{array}{*{20}{l}}{{{\left( { – \frac{1}{3}} \right)}^{ – 4}} \times {3^{ – 4}}}& = &{{{\left( { – \frac{1}{3} \times 3} \right)}^{ – 4}}}\\{}& = &{{{\left( { – 1} \right)}^{ – 4}}}\\{}& = &{{{\left( { – 1} \right)}^4}}\\{}& = &1\end{array}\] | \[\begin{array}{*{20}{l}}{{2^2} \times {{\left( { – 2} \right)}^3}}& = &{{{\left( { – 2} \right)}^2} \times {{\left( { – 2} \right)}^3}}\\{}& = &{{{\left( { – 2} \right)}^{2 + 3}}}\\{}& = &{{{\left( { – 2} \right)}^5}}\\{}& = &{ – 32}\end{array}\] |
| \[\begin{array}{*{20}{l}}{ – \frac{2}{3} \times {{\left( {\frac{3}{2}} \right)}^{ – 3}}}& = &{ – \left[ {\frac{2}{3} \times {{\left( {\frac{2}{3}} \right)}^{ + 3}}} \right]}\\{}& = &{ – {{\left( {\frac{2}{3}} \right)}^{1 + 3}}}\\{}& = &{ – {{\left( {\frac{2}{3}} \right)}^4}}\\{}& = &{ – \frac{{16}}{{81}}}\end{array}\] | \[\begin{array}{*{20}{l}}{{{\left( { – 4} \right)}^6} \div {2^6}}& = &{{{\left( { – 4 \div 2} \right)}^6}}\\{}& = &{{{\left( { – 2} \right)}^6}}\\{}& = &{64}\end{array}\] | \[\begin{array}{*{20}{l}}{\left( { – \frac{1}{2}} \right) \div {{\left( { – \frac{1}{3}} \right)}^4}}& = &{ – \frac{1}{2} \div \frac{1}{{81}}}\\{}& = &{ – \frac{1}{2} \times \frac{{81}}{1}}\\{}& = &{ – \frac{{81}}{2}}\end{array}\] |
| \[\begin{array}{*{20}{l}}{{2^3} \times {{\left( { – 2} \right)}^{ – 4}}}& = &{{2^3} \times {{\left( { + 2} \right)}^{ – 4}}}\\{}& = &{{2^{3 + \left( { – 4} \right)}}}\\{}& = &{{2^{ – 1}}}\\{}& = &{\frac{1}{2}}\end{array}\] | \[\begin{array}{*{20}{l}}{{{\left( { – 3} \right)}^5} \div {3^5}}& = &{ – \left( {{{\left( { + 3} \right)}^5} \div {3^5}} \right)}\\{}& = &{ – \left( {{3^{5 – 5}}} \right)}\\{}& = &{ – {3^0}}\\{}& = &{ – 1}\end{array}\] | \[\begin{array}{*{20}{l}}{{{\left( { – 1} \right)}^{102}}}& = &{{1^{102}}}\\{}& = &1\end{array}\] |
| \[\begin{array}{*{20}{l}}{{2^3} + {2^4}}& = &{8 + 16}\\{}& = &{24}\end{array}\] | \[\begin{array}{*{20}{l}}{{3^{ – 2}} – {{\left( { – 3} \right)}^{ – 2}}}& = &{{{\left( {\frac{1}{3}} \right)}^2} – {{\left( { – \frac{1}{3}} \right)}^2}}\\{}& = &{\frac{1}{9} – \frac{1}{9}}\\{}& = &0\end{array}\] | \[\begin{array}{*{20}{l}}{{{\left( { – 2} \right)}^2} + {{\left( { – 3} \right)}^2}}& = &{4 + 9}\\{}& = &{13}\end{array}\] |
