a) \(\dfrac{1}{10}\)
b) \(\dfrac{1}{15}\)
c) \(\dfrac{1}{10} + \dfrac{1}{15} = \dfrac{1}{6}\)
d) 6 minutos
a) O menor número é 0,90 e o maior é 10,01.
b) \(10,01 - 0,90 = 9,11\).
Se \(\dfrac{3}{10}\) dos 2940 alunos jogam futebol, podemos calcular isso assim:
\[ 2940 \times \frac{3}{10} = 882\]
Agora, temos que \(\dfrac{2}{7}\) dos 2940 alunos fazem natação, então:
\[ 2940 \times \frac{2}{7} = 840\]
Agora, somando esses números, temos: \(882 + 840 = 1722\), ou seja, \(2940 - 1722 = 1218\) alunos não praticam esportes.
\[ \begin{array}{c} - \left\lbrace - \left[ - \left( 1 - \dfrac{1}{3} - \dfrac{3}{4} \right) \right] \right\rbrace\\ - \left\lbrace - \left[ - \left( \dfrac{12}{12} - \dfrac{4}{12} - \dfrac{9}{12} \right) \right] \right\rbrace\\ - \left\lbrace - \left[ - \left(- \dfrac{1}{12} \right) \right] \right\rbrace\\ + \dfrac{1}{12} \end{array}\]
\[ \begin{array}{c} \dfrac{1}{6} - \left[ 1,1 - \left( \dfrac{1}{2} + 0,4 \right) \right]\\ \dfrac{1}{6} - \left[ \dfrac{11}{10} - \left( \dfrac{1}{2} + \dfrac{4}{10} \right) \right]\\ \dfrac{5}{30} - \left[ \dfrac{33}{30} - \left( \dfrac{15}{30} + \dfrac{12}{30} \right)\right]\\ \dfrac{5}{30} - \left[ \dfrac{33}{30} - \dfrac{27}{30} \right]\\ \dfrac{5}{30} - \dfrac{6}{30}\\ -\dfrac{1}{30} \end{array}\]
Está entre 0 e −1.
a) Se ela guardou \(\dfrac{7}{9}\) e usará \(\dfrac{4}{5}\), restará \(\dfrac{1}{5}\). Então basta fazer \(\dfrac{7}{9} \times \dfrac{1}{5} = \dfrac{7}{45}\).
b) Se ela tivesse usado \(\dfrac{1}{2}\), restaria \(\dfrac{1}{2}\). Então fazemos \(\dfrac{7}{9} \times \dfrac{1}{2} = \dfrac{7}{18}\).
a) \(\left( \dfrac{1}{3} + \dfrac{1}{9} \right) \cdot \left( \dfrac{5}{4} - \dfrac{3}{4} \right)\)
\[ \left( \dfrac{3}{9} + \dfrac{1}{9} \right) \cdot \left( \dfrac{2}{4} \right)\\ \left( \dfrac{4}{9} \right) \cdot \left( \dfrac{2}{4} \right)\\ \dfrac{8}{36}\\ \dfrac{2}{9}\]
b) \(\left[ \left( \dfrac{1}{5} + 1 \right) \cdot 5 \right] + \dfrac{1}{3}\)
\[ \left[ \left( \dfrac{1}{5} + \dfrac{5}{5} \right) \cdot 5 \right] + \dfrac{1}{3}\\ \left[ \left( \dfrac{6}{5} \right) \cdot 5 \right] + \dfrac{1}{3}\\ \dfrac{30}{5} + \dfrac{1}{3}\\ 6 + \dfrac{1}{3}\\ \dfrac{18}{3} + \dfrac{1}{3}\\ \dfrac{19}{3}\]
c) \(\left[ \left( \dfrac{7}{5} - 2 \right) \cdot 2 \right] + \dfrac{1}{5}\)
\[ \left[ \left( \dfrac{7}{5} - \dfrac{10}{5} \right) \cdot 2 \right] + \dfrac{1}{5}\\ \left[ \left( - \dfrac{3}{5} \right) \cdot 2 \right] + \dfrac{1}{5}\\ \left[ - \dfrac{6}{5} \right] + \dfrac{1}{5}\\ - \dfrac{5}{5}\\ - 1\]
d) \(\left[ \left( 0,5 \cdot 2 + \dfrac{1}{3} \right) \cdot \dfrac{2}{5} \right]\)
\[ \left[ \left( 1 + \dfrac{1}{3} \right) \cdot \dfrac{2}{5} \right]\\ \left[ \left( \dfrac{3}{3} + \dfrac{1}{3} \right) \cdot \dfrac{2}{5} \right]\\ \left[ \dfrac{4}{3} \cdot \dfrac{2}{5} \right]\\ \dfrac{8}{15}\]
e) \(\left[ \left( \dfrac{2}{5} - \dfrac{3}{4} \right) \cdot \dfrac{2}{3} \right] \div 3\)
\[ \left[ \left( \dfrac{8}{20} - \dfrac{15}{20} \right) \cdot \dfrac{2}{3} \right] \div 3\\ \left[ \left( - \dfrac{7}{20} \right) \cdot \dfrac{2}{3} \right] \div 3\\ \left[ - \dfrac{14}{60} \right] \div 3\\ - \dfrac{14}{60} \times \dfrac{1}{3}\\ - \dfrac{14}{180} \\ - \dfrac{7}{90} \]
f) \(\left[ \left( \dfrac{25}{10} - \dfrac{3}{5} \right) \times 4 \right] \div 2\)
\[ \left[ \left( \dfrac{25}{10} - \dfrac{6}{10} \right) \times 4 \right] \div 2\\ \left[ \dfrac{19}{10} \times 4 \right] \div 2\\ \dfrac{76}{10} \div 2\\ \dfrac{76}{10} \times \dfrac{1}{2}\\ \dfrac{76}{20}\\ \dfrac{19}{5}\]
g) \(\left[ 5 \times \left( \dfrac{1}{3} + \dfrac{1}{9} \right) \times \left( \dfrac{5}{4} - \dfrac{3}{4} \right) + \dfrac{1}{3} \right] \div \dfrac{1}{2}\)
\[ \left[ 5 \times \left( \dfrac{3}{9} + \dfrac{1}{9} \right) \times \dfrac{2}{4} + \dfrac{1}{3} \right] \div \dfrac{1}{2}\\ \left[ 5 \times \dfrac{4}{9} \times \dfrac{2}{4} + \dfrac{1}{3} \right] \div \dfrac{1}{2}\\ \left[ \dfrac{40}{36} + \dfrac{1}{3} \right] \div \dfrac{1}{2}\\ \left[ \dfrac{10}{9} + \dfrac{1}{3} \right] \div \dfrac{1}{2}\\ \left[ \dfrac{10}{9} + \dfrac{3}{9} \right] \div \dfrac{1}{2}\\ \dfrac{13}{9} \div \dfrac{1}{2}\\ \dfrac{13}{9} \times \dfrac{2}{1}\\ \dfrac{26}{9}\]
h) \(\left\{ \left[ \left( \dfrac{1}{2} + \dfrac{1}{3} \right) + \left( \dfrac{1}{3} - \dfrac{1}{4} \right) \right] \times \dfrac{1}{3} \right\} \times \dfrac{1}{2}\)
\[ \left\{ \left[ \left( \dfrac{3}{6} + \dfrac{2}{6} \right) + \left( \dfrac{4}{12} - \dfrac{3}{12} \right) \right] \times \dfrac{1}{3} \right\} \times \dfrac{1}{2}\\ \left\{ \left[ \dfrac{5}{6} + \dfrac{1}{12} \right] \times \dfrac{1}{3} \right\} \times \dfrac{1}{2}\\ \left\{ \left[ \dfrac{10}{12} + \dfrac{1}{12} \right] \times \dfrac{1}{3} \right\} \times \dfrac{1}{2}\\ \left\{ \dfrac{11}{12} \times \dfrac{1}{3} \right\} \times \dfrac{1}{2}\\ \dfrac{11}{36} \times \dfrac{1}{2}\\ \dfrac{11}{72}\]
Alternativa E
a) \(\left( \dfrac{1}{2} \right)^{4} + \left( \dfrac{1}{2} \right)^{2} \neq \left( \dfrac{1}{2}\right)^{6}\)
b) \(\left[ \dfrac{1}{3} \cdot \left( \dfrac{2}{5} \right) \right]^{2} = \left( \dfrac{1}{3} \right)^{2} \cdot \left( \dfrac{2}{5} \right)^{2}\)
c) \(\left( \dfrac{1}{3} + \dfrac{2}{5} \right) \neq \left( \dfrac{1}{3} \right)^{2} + \left( \dfrac{2}{5} \right)^{2}\)
d) \(\left( \dfrac{1}{3} + \dfrac{2}{5} \right)^{2} = \left( \dfrac{1}{3} \right)^{2} + \dfrac{4}{15} + \left( \dfrac{2}{5} \right)^{2}\)
a) \((0,4)^{2} \cdot (0,4)^{4} = (0,4)^{6}\)
b) \(\left( \dfrac{2}{3} \right)^{3} \div \left( \dfrac{2}{3} \right)^{2} = \left( \dfrac{2}{3} \right)\)
c) \(\left[ \left( - \dfrac{2}{3} \right)^{2} \right]^{3} = \left( - \dfrac{2}{3} \right)^{6}\)
d) \(\left( - \dfrac{1}{5} \right)^{2} \cdot \left( - \dfrac{1}{5} \right)^{3} = \left( -\dfrac{1}{5} \right)^{-1} = (-5)\)
e) \(\left( -\dfrac{1}{2} \right)^{-4} \cdot \left( - \dfrac{1}{2} \right)^{-2} = \left( -\dfrac{1}{2} \right)^{-6} = (-2)^{6}\)
f) \(\left[ \left( \dfrac{2}{5} \right)^{2} \right]^{5} = \left( \dfrac{2}{5} \right)^{10}\)
a) \(\left( \dfrac{2}{5} \right)^{3} \cdot \left( \dfrac{2}{5} \right)^{4} \cdot \left( \dfrac{2}{5} \right)^{6} = \left( \dfrac{2}{5} \right)^{13}\)
b) \(\left( - \dfrac{1}{5} \right)^{8} \div \left( - \dfrac{1}{5} \right)^{1} = \left( -\dfrac{1}{5} \right)^{7}\)
c) \(\left[ \left( - \dfrac{2}{15} \right)^{8} \right]^{-2} = \left( - \dfrac{2}{15} \right)^{-16} = \left( - \dfrac{15}{2} \right)^{16}\)
d) \(\left[ \left( \dfrac{1}{2} \right)^{5} \cdot \left( \dfrac{1}{2} \right)^{-3}\right] \div \left( \dfrac{1}{2} \right)^{-5} = \left( \dfrac{1}{2} \right)^{2} \div \left( \dfrac{1}{2} \right)^{-5} = \left( \dfrac{1}{2} \right)^{7}\)
e) \(\left[ \left( - \dfrac{8}{7} \right)^{-3} \div \left( - \dfrac{8}{7} \right)^{-10} \right] \cdot \left( -\dfrac{8}{7} \right)^{-18} = \left( -\dfrac{8}{7} \right)^{7} \cdot \left( - \dfrac{8}{7} \right)^{-18} = \left( -\dfrac{8}{7} \right)^{-11} = \left( \dfrac{7}{8} \right)^{11}\)
f) \(\left[ \left( \dfrac{7}{4} + \dfrac{2}{4} \right)^{2} \right]^{0} = 1\)
a) \(\dfrac{2}{5} + \left( - \dfrac{2}{7} \right) \div \left( \dfrac{5}{14} \right)\)
\[ \begin{array}{c} \dfrac{2}{5} + \left( - \dfrac{2}{7} \right) \times \left( \dfrac{14}{5} \right)\\ \dfrac{2}{5} + \left( - \dfrac{4}{5} \right)\\ - \dfrac{2}{5} \end{array}\]
b) \(\left[ \left( - \dfrac{1}{5} \right) \div \left( -\dfrac{1}{7} \right) - \left( -\dfrac{2}{11} \right) \div \left( \dfrac{4}{5} \right) \right]\)
\[ \begin{array}{c} \left[ \left( - \dfrac{1}{5} \right) \times \left( -\dfrac{7}{1} \right) - \left( -\dfrac{2}{11} \right) \times \left( \dfrac{5}{4} \right) \right]\\ \left[ \left( \dfrac{7}{5} \right) - \left( -\dfrac{5}{22} \right) \right]\\ \dfrac{154}{110} +\dfrac{25}{110} \\ \dfrac{179}{110}\\ \end{array}\]
c) \(\left[ 3 \cdot \left( - \dfrac{1}{3} \right)^{6} + 4 \cdot \dfrac{5}{4} \right]^{0} = 1\)
d) \(\dfrac{5}{4} - \sqrt{\dfrac{16}{9}} + \sqrt{\dfrac{4}{9}}\)
\[ \begin{array}{c} \dfrac{5}{4} - \dfrac{4}{3} + \dfrac{2}{3}\\ \dfrac{15}{12} - \dfrac{16}{12} + \dfrac{8}{12}\\ \dfrac{7}{12} \end{array}\]
e) \(\sqrt{\dfrac{25}{9} \cdot \dfrac{16}{9} } \cdot \dfrac{6}{3}\)
\[ \begin{array}{c} \sqrt{\dfrac{400}{81} } \cdot \dfrac{6}{3}\\ \dfrac{20}{9} \cdot \dfrac{6}{3}\\ \dfrac{120}{27}\\ \dfrac{40}{9} \end{array}\]
f) \(\left[ \left( \sqrt{\dfrac{68}{13} - 6} \right) - \left( \dfrac{75}{12} \right) \right]^{0} = 1\)
a) \(\left( \dfrac{2}{3} \right)^{2} = \dfrac{4}{9}\)
b) \(\left( \dfrac{2}{3} \right)^{3} = \dfrac{8}{27}\)
c) \(\left( -\dfrac{2}{3} \right)^{4} = \dfrac{16}{81}\)
d) \(\left( -\dfrac{2}{3} \right)^{5} = -\dfrac{32}{243}\)
e) \(\left( \dfrac{1}{2} \right)^{6} = \dfrac{1}{64}\)
f) \(\left( \dfrac{1}{3} \right)^{2} \div \left( \dfrac{1}{3} \right) = \dfrac{1}{3}\)
g) \(\left( \dfrac{12}{9} \right)^{2} = \dfrac{144}{81} = \dfrac{16}{9}\)
h) \(\left( \dfrac{2}{4} \right) \cdot \left( \dfrac{2}{4} \right) = \dfrac{4}{16} = \dfrac{1}{4}\)
i) \(\left( \dfrac{1}{2} \right)^{3} + \left( \dfrac{1}{4} \right) =\dfrac{1}{8} + \dfrac{1}{4} = \dfrac{1}{8} + \dfrac{2}{8} = \dfrac{3}{8}\)
j) \(\left( \dfrac{1}{20} \right)^{1} + \left( \dfrac{1}{2} \right)^{2} = \dfrac{1}{20} + \dfrac{1}{4} = \dfrac{3}{10}\)
k) \(\left( \dfrac{1}{8} \right)^{2} - \left( \dfrac{1}{4} \right)^{2} = \dfrac{1}{64} - \dfrac{1}{16} = -\dfrac{3}{64}\)
l) \(\left( \dfrac{182}{180} \right)^{0} + \left( \dfrac{211}{230} \right)^{1} = 1 + \dfrac{211}{230} = \dfrac{441}{230}\)
Para estes últimos exercícios apresentaremos apenas as respostas:
a) \(\dfrac{1}{3}\)
b) \(\dfrac{1}{2}\)
c) \(\dfrac{11}{8}\)
d) 2
e) \(\dfrac{9}{4}\)
f) \(\dfrac{1}{10}\)
g) \(\dfrac{1}{2}\)
h) \(\dfrac{1}{3}\)
i) \(\dfrac{1}{5}\)
j) 1
k) 0
l) \(\dfrac{13}{6}\)
a) \(\dfrac{13}{50}\)
b) \(\dfrac{407}{500}\)
c) \(\dfrac{343}{500}\)
d) \(\dfrac{43}{24}\)
e) \(\dfrac{148}{45}\)
f) \(\dfrac{59}{90}\)
a) \(\left( \dfrac{4}{3} \right)^{4}\)
b) \(\left( -\dfrac{3}{10} \right)^{18}\)
c) \(\left( \dfrac{2}{3} \right)^{5}\)
d) \(\left( -\dfrac{7}{9} \right)^{-15} = \left( -\dfrac{9}{7} \right)^{15}\)