Odpowiedź:
zad. 1
1.
[tex] {9}^{7} = ({3}^{2} {)}^{7} = {3}^{14} \\ {3}^{14} \div {3}^{7} = {3}^{7} [/tex]
nie
2.
[tex]( {3}^{4} {)}^{5} = {3}^{20} \\ {3}^{20} \div {3}^{19} = {3}^{1} = 3[/tex]
tak
3.
[tex] {9}^{5} \div {3}^{9} = ({3}^{2} {)}^{5} \div {3}^{9} = {3}^{10} \div {3}^{9} = {3}^{1} = 3[/tex]
tak
4.
[tex] \frac{ {3}^{15} - {3}^{13} }{ { 3}^{12} } = \frac{( {3}^{2} - 3) {3}^{13} }{ {3}^{12} } = (9 - 1) \times 3 = 8 \times 3 = 24[/tex]
nie
zad. 2
[tex] \frac{1}{2} \times {2}^{30} = {2}^{ - 1} \times {2}^{30} = {2}^{29} [/tex]
zad. 3
[tex] {36}^{3} \times ( \frac{1}{3} {)}^{6} = ({6}^{2} {)}^{3} \times {3}^{ - 6} = {6}^{6} \times {3}^{ - 6} = {2}^{6} \times {3}^{6} \times {3}^{ - 6} = {2}^{6} [/tex]
zad. 4
[tex] {4}^{3} + {2}^{5} + {4}^{3} + {2}^{5} + {4}^{3} = ( {2}^{2} {)}^{3} + {2}^{5} + ( {2}^{2} {)}^{3} + {2}^{5} + ({2}^{2} {)}^{3} = {2}^{6} + {2}^{5} + {2}^{6} + {2}^{5} + {2}^{6} = {2}^{6} + 2 \times {2}^{5} + {2}^{6} + 2 {}^{6} = 3 \times {2}^{6} + {2}^{6} = 4 \times {2}^{6} = {2}^{2} \times {2}^{6} = {2}^{8} = 256[/tex]
zad. 5
[tex] \frac{ {3}^{4} \times {9}^{6} }{ {243}^{5} \div {27}^{4} } = \frac{ {3}^{4} \times ({3}^{2} {)}^{6} }{( {3}^{5} {)}^{5} \div ( {3}^{3} {)}^{4} } = \frac{ {3}^{4} \times {3}^{12} }{ {3}^{25} \div {3}^{12} } = \frac{ {3}^{16} }{ {3}^{13} } = {3}^{3} = 27[/tex]
zad. 6
[tex] \frac{( {3}^{3} {)}^{ - 2} \times {27}^{14} }{ {81}^{ - 2} \div (3 \sqrt{3} {)}^{6} } = \frac{ {3}^{ - 6} \times ( {3}^{3} {)}^{14} }{( {3}^{4} {)}^{ - 2} \div ( {3}^{ \frac{3}{2} } {)}^{6} } = \frac{ {3}^{ - 6} \times {3}^{42} }{ {3}^{ - 8} \div {3}^{9} } = \frac{ {3}^{36} }{ {3}^{ - 17} } = {3}^{53} [/tex]
