[tex]1) \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ 128 \times {( \frac{1}{8}) }^{2} = 128 \times \frac{1}{64} = 2 \\ { ({6}^{3}) }^{2} \div {36}^{5} = {6}^{6} \div {( {6}^{2}) }^{5} = {6}^{6} \div {6}^{7} = {6}^{ - 1} = \frac{1}{6} \\ \\ 128 \times( \frac{1}{8} ) {}^{2} > { ({6}^{3} )}^{2} \div {6}^{5}[/tex]
[tex]2) \: \: \: \: \: \frac{ {8}^{3} \times {2}^{5} }{ {16}^{4} } < 1 \frac{1}{8} [/tex]
[tex]\frac{ {8}^{3} \times {2}^{5} }{ {16}^{4} } = \frac{ ({2 {}^{3} )}^{3} \times {2}^{5} }{( { {2}^{4}) }^{4} } = \frac{ {2}^{9} \times {2}^{5} }{ {2}^{16} } = \frac{ {2}^{14} }{ {2}^{16} } = \frac{1}{ {2}^{2} } = \frac{1}{4} = 0.25[/tex]
[tex]1 \frac{1}{8} = 1.125[[/tex]
[tex]3)13 \times {4}^{3} \div {2}^{3} > {10}^{2} \div {5}^{2} \div {2}^{3} [/tex]
[tex]{10}^{2} \div {5}^{2} \div {2}^{3} = {2}^{2} \times {5}^{2} \div {5}^{2} \div {2}^{3} = {2}^{ - 1} = \frac{1}{2} = 0.5[/tex]
[tex]4) \frac{14 \times {3}^{2} \div {4}^{2} }{2 \times {3}^{3} } < \frac{ {21}^{3} \times 5 }{ {7}^{3} \times {3}^{4} } [/tex]
[tex] \frac{14 \times {3}^{2} \div {4}^{2} }{2 \times {3}^{3} } = \frac{2 \times 7 \times {3}^{2} }{2 \times {3}^{3} \times {( {2}^{2}) }^{2} } = \frac{2 \times 7 \times {3}^{2} }{ {2}^{5} \times {3}^{3} } = \frac{7}{ {2}^{4} \times 3 } = \frac{7}{48} [/tex]
[tex] \frac{ {21}^{3} \times 5 }{ {7}^{3} \times {3}^{4} } = \frac{ {7}^{3} \times {3}^{3} \times 5 }{ {7}^{3} \times {3}^{4} } = \frac{5}{3} = 1\frac{2}{3} [/tex]
[tex]5)( - 0.5)^{3} \times 16 + 4 < {2}^{3} - 2.6[/tex]
[tex]( - 0.5)^{3} \times 16 + 4 = - 0.125 \times 16 + 4 = - 2 + 4 = 2[/tex]
[tex] {2}^{3} - 2.6 = 8 - 2.6 = 5.4[/tex]
[tex]6) {( - \frac{1}{4}) }^{4} \times 243 + {6}^{3} - 64 < {6}^{3} + ( { - 3)}^{2} - {2}^{5} [/tex]
[tex]( - \frac{1}{3} {)}^{4} \times 243 + {6}^{3} - 64 = \frac{1}{81} \times 243 + 216 - 64 = 3 + 216 - 64 = 219 - 64 = 155[/tex]
[tex] {6}^{3} + ( { - 3)}^{2} - {2}^{5} = 216 + 9 - 32 = 225 - 32 = 193[/tex]
[tex]7) {50}^{4} > {2}^{4} \div {5}^{ - 6} [/tex]
[tex] {50}^{4} = {25}^{4} \times {2}^{4} = { ({5}^{2} )}^{4} \times {2}^{4} = {5}^{8} \times {2}^{4} [/tex]
[tex]{2}^{4} \div {5}^{ - 6} = {2}^{4} \div \frac{1}{ {5}^{6} } = {2}^{4} \times {5}^{6}[/tex]
[tex]8) {90}^{4} < {3}^{4} \div {10}^{ - 6} [/tex]
[tex] {90}^{4} = {9}^{4} \times {10}^{4} = { ({3}^{2} )}^{4} \times {10}^{4} = {3}^{8} \times {10}^{4} [/tex]
[tex] {3}^{4} \div {10}^{ - 6} = {3}^{4} \div \frac{1}{10 {}^{6} } = {3}^{4} \times {10}^{6} [/tex]
