Calcula o valor das expressões numéricas
Números reais: Matematicamente Falando 8 - Pág. 12 Ex. 7
Enunciado
Calcula o valor das expressões numéricas utilizando, sempre que possível, as regras de operações com potências:
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Resolução
- \({\left( {{3^2}} \right)^{ – 1}} = {3^{2 \times \left( { – 1} \right)}} = {3^{ – 2}} = {\left( {\frac{1}{3}} \right)^2} = \frac{1}{9}\)
- \({\left[ {{{\left( {\frac{3}{2}} \right)}^2}} \right]^0} = {\left( {\frac{3}{2}} \right)^{2 \times 0}} = {\left( {\frac{3}{2}} \right)^0} = 1\)
- \({\left[ {{{\left( { – 5} \right)}^2}} \right]^{ – 2}} = {\left( { – 5} \right)^{ – 4}} = {\left( { – \frac{1}{5}} \right)^4} = \frac{1}{{625}}\)
- \({\left[ {{{\left( { – 0,25} \right)}^3}} \right]^0} = {\left( { – 0,25} \right)^0} = 1\)
- \({7^2} \times {7^{ – 3}} \times {7^{ – 1}} = {7^{2 + \left( { – 3} \right) + \left( { – 1} \right)}} = {7^{ – 2}} = {\left( {\frac{1}{7}} \right)^2} = \frac{1}{{49}}\)
- \({\left( { – 2} \right)^3} \times {3^3} = {\left( { – 2 \times 3} \right)^3} = {\left( { – 6} \right)^3} = – 216\)
- \({\left( { – 2} \right)^3} \times {\left( { – 2} \right)^5} \div {\left( { – 2} \right)^{ – 3}} = {\left( { – 2} \right)^8} \div {\left( { – 2} \right)^{ – 3}} = {\left( { – 2} \right)^{8 – \left( { – 3} \right)}} = {\left( { – 2} \right)^{11}} = – 2 \times {2^{10}} = – 2 \times 1024 = – 2048\)
- \({\left( { – 0,5} \right)^{ – 3}} \times {\left( { – 0,5} \right)^{ – 2}} = {\left( { – 0,5} \right)^{ – 3 + \left( { – 2} \right)}} = {\left( { – \frac{1}{2}} \right)^{ – 5}} = {\left( { – 2} \right)^5} = – 32\)
- \({5^{ – 4}} \div {5^3} \times {\left( { – \frac{1}{5}} \right)^{ – 7}} = {5^{ – 7}} \times {\left( { – \frac{1}{5}} \right)^{ – 7}} = {\left( {5 \times \left( { – \frac{1}{5}} \right)} \right)^{ – 7}} = {\left( { – 1} \right)^{ – 7}} = {\left( { – 1} \right)^7} = – 1\)
- \(\left( {2 \times {2^4}} \right) \div \left( {{2^{10}} \times {2^{ – 3}}} \right) = {2^5} \div {2^7} = {2^{ – 2}} = {\left( {\frac{1}{2}} \right)^2} = \frac{1}{4}\)
- \(\frac{5}{4} \div {\left( {\frac{5}{4}} \right)^{ – 2}} = {\left( {\frac{5}{4}} \right)^1} \div {\left( {\frac{5}{4}} \right)^{ – 2}} = {\left( {\frac{5}{4}} \right)^{1 – \left( { – 2} \right)}} = {\left( {\frac{5}{4}} \right)^3} = \frac{{125}}{{64}}\)
- \({\left( { – \frac{1}{7}} \right)^{ – 2}} \times {7^{ – 2}} = {\left( { – \frac{1}{7} \times 7} \right)^{ – 2}} = {\left( { – 1} \right)^{ – 2}} = {\left( { – 1} \right)^2} = 1\)
- \({\left( { – \frac{1}{2}} \right)^0} \div {\left( { – \frac{1}{2}} \right)^{ – 4}} \times {\left( { – \frac{1}{2}} \right)^{ – 2}} = {\left( { – \frac{1}{2}} \right)^{0 – \left( { – 4} \right)}} \times {\left( { – \frac{1}{2}} \right)^{ – 2}} = {\left( { – \frac{1}{2}} \right)^4} \times {\left( { – \frac{1}{2}} \right)^{ – 2}} = {\left( { – \frac{1}{2}} \right)^2} = \frac{1}{4}\)
- \({\left( {1 + \frac{1}{3}} \right)^{ – 7}} \times {\left( {\frac{8}{3}} \right)^7} \div {2^4} = {\left( {\frac{4}{3}} \right)^{ – 7}} \times {\left( {\frac{8}{3}} \right)^7} \div {2^4} = {\left( {\frac{3}{4}} \right)^7} \times {\left( {\frac{8}{3}} \right)^7} \div {2^4} = {\left( {\frac{3}{4} \times \frac{8}{3}} \right)^7} \div {2^4} = {2^7} \div {2^4} = {2^3} = 8\)
- \({\left( { – \frac{1}{2}} \right)^3} \div {\left( { – \frac{1}{2}} \right)^2} + {\left( { – \frac{3}{5}} \right)^4} \div {\left( { – \frac{3}{5}} \right)^6} = {\left( { – \frac{1}{2}} \right)^1} + {\left( { – \frac{3}{5}} \right)^{ – 2}} = – \frac{1}{2} + {\left( { – \frac{5}{3}} \right)^2} = – \frac{1}{2} + \frac{{25}}{9} = – \frac{9}{{18}} + \frac{{50}}{{18}} = \frac{{41}}{{18}}\)
