درسنامه کامل ریاضی هشتم فصل 7 توان و جذر

\(\begin{array}{l}1)\,{10^ \circ } = 1\;\;\;\;\;\;\;\\\\2){1^7} = 7\;\\\\3){ \circ ^4} = \; \circ \\\\4){5^1} = 5\\\\5){( - 6)^2} = \,( - 6) \times ( - 6) = 36\\\\6) - {3^3} = - (3 \times 3 \times 3) = - 27\\\\7){( - 2)^ \circ } = 1\\\,\,\\8)( - 3) \times ( - 3) \times ( - 3) = {( - 3)^2} = - 27\\\\9){11^2} = 11 \times 11 = 121\,\,\,\,\,\\\\10){6^1} = 6\\\\11){ \circ ^1} = \circ \\\\12){( - 2)^ \circ } \times {2^5} = 1 \times 2 \times 2 \times 2 \times 2 \times 2 = 32\end{array}\)

\(\begin{array}{l}1)\,{( - 1)^4} = 1\\\\2){( - 1)^3} = - 1\\\\3)( - {3^4}) = - (3 \times 3 \times 3 \times 3) = - 81\\\\4) - {5^2} = - (5 \times 5) = - 25\\\\5)\frac{4}{{{3^2}}} = \frac{4}{{3 \times 3}} = \frac{4}{9}\\\\6)\frac{{{{( - 4)}^2}}}{{{4^2}}} = \frac{{( - 4) \times ( - 4)}}{{4 \times 4}} = \frac{{16}}{{16}} = 1\\\\7){( - \frac{2}{5})^2} = ( - \frac{2}{5}) \times ( - \frac{2}{5}) = \frac{4}{{25}}\\\\8){(\frac{a}{b})^3} = \frac{a}{b} \times \frac{a}{b} \times \frac{a}{b} = \frac{{a \times a \times a}}{{b \times b \times b}} = \frac{{{a^3}}}{{{b^3}}}\end{array}\)

\(\frac{{{5^3} - {3^3}}}{{{5^3} + {3^3}}} = \frac{{125 - 27}}{{125 + 27}} = \frac{{98}}{{152}} = \frac{{49}}{{76}}\)

\(\begin{array}{l}58906 = 50000 + 8000 + 900 + 6\\\\\,\,\,\,\,\,\,\,\,\,\,\,\, = 5 \times {10^4} + 8 \times {10^3} + 9 \times {10^2} + \circ \times {10^1} + 6 \times {10^ \circ }\\\\40200 = 40000 + 200\\\,\,\,\,\,\,\,\,\,\,\,\,\, = 4 \times {10^4} + \circ \times {10^3} + 2 \times {10^2} + \circ \times {10^1} + \circ \times {10^ \circ }\\\end{array}\)

\({2^{12}} = {2^{10}} \times {2^2} = 1024 \times 4 = 4096\)

\({12^7} = {(2 \times 6)^7} = {2^7} \times {6^7}\)

\(\begin{array}{l}\underline {{5^2} \times {5^7}} \times {7^9} = {5^{2 + 7}} \times {7^9} = {5^9} \times {7^9} = {(5 \times 7)^9}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {35^9}\end{array}\)

\({2^{a + 1}} = {2^a} \times {2^1} = 7 \times 2 = 14\)

1) فقط به اولویت عملیاتی دقت کنید:

\(\begin{array}{l} - {(1 - 2(1 - 2(1 - 2(1 - {2^{ - 1}}))))^{ - 1}}\mathop = \limits^{{2^{ - 1}} = \frac{1}{2}} \\\\ - {(1 - 2(1 - 2(1 - 2(1 - \frac{1}{2}))))^{ - 1}} = \\\\ - {(1 - 2(1 - 2(1 - 2(\frac{1}{2}))))^{ - 1}} = \\\\ - {(1 - 2(1 - 2(1 - 0)))^{ - 1}} = \\\\ - {(1 - 2(1 - 2))^{ - 1}} = \\\\ - {(1 - 2( - 1))^{ - 1}} = - {(1 + 2)^{ - 1}} = - {(3)^{ - 1}} = - \frac{1}{3}\end{array}\)

2)

\(\begin{array}{l}\frac{{{4^{a + 2}} - {4^{a + 1}} - {4^a}}}{{{2^a} + {2^a} + {2^a} + {2^a}}} = \frac{{{4^a}({4^2} - 4 - 1)}}{{{2^a} \times 4}} = \\\\\frac{{{4^a}}}{{{2^a}}} \times \frac{{11}}{4} = {(\frac{4}{2})^a} \times \frac{{11}}{4} = {2^a} \times \frac{{11}}{4}\end{array}\)

3)

\(\begin{array}{l}\,\frac{{{2^{101}} + {2^{100}} + {2^{99}} \cdots + {2^{60}}}}{{{2^{61}} + {2^{60}} + {2^{59}} \cdots + {2^{10}}}} = \\\\\frac{{{2^{60}}({2^{51}} + {2^{50}} + {2^{49}} \cdots + 2 + 1)}}{{{2^{10}}({2^{51}} + {2^{50}} + {2^{49}} \cdots + 2 + 1)}} = \frac{{{2^{60}}}}{{{2^{10}}}} = {2^{60 - 10}} = {2^{50}}\\\end{array}\)

4)

\(\,\frac{{{3^{2a + 2}} \div {3^{2a - 2}}}}{{{9^{2b + 2}} \times {9^{ - 2b}}}} = \frac{{{3^{(2a + 2) - (2a - 2)}}}}{{{9^{(2b + 2 + ( - 2b))}}}} = \frac{{{3^4}}}{{{9^2}}} = \frac{{81}}{{81}} = 1\)

\(\begin{array}{l}1)\,\sqrt {49} = 7\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2)\,\sqrt {0/25} = 0/5\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\\\3)\,\sqrt {8/1} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,4)\,\sqrt {8 \times 2} \, = \sqrt {16} = 4\\\\5)\,\sqrt 3 \times \sqrt {27} = \sqrt {3 \times 27} = \sqrt {81} = 9\\\\6)\,\sqrt {1\,} = 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,7)\sqrt {125} \,\\\\8)\sqrt {169} = 13\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,9)\,\sqrt {\frac{{10}}{{0/1}}} = \sqrt {10 \times 10} = 10\end{array}\)

\(\sqrt {\sqrt {\sqrt {\sqrt {{{81}^2}} } } } = \sqrt {\sqrt {\sqrt {81} } } = \sqrt {\sqrt 9 } = \sqrt 3 \)

\(\begin{array}{l}1)\sqrt {2\sqrt {2\sqrt {2\sqrt {2\sqrt {2\sqrt {2\sqrt {2\sqrt {2\sqrt 4 } } } } } } } } = \\\sqrt {2\sqrt {2\sqrt {2\sqrt {2\sqrt {2\sqrt {2\sqrt {2\sqrt {2 \times 2} } } } } } } } = \\\sqrt {2\sqrt {2\sqrt {2\sqrt {2\sqrt {2\sqrt {2\sqrt 4 } } } } } } = \sqrt {2\sqrt {2\sqrt {2\sqrt {2\sqrt {2\sqrt {2 \times 2} } } } } } = \\\sqrt {2\sqrt {2\sqrt {2\sqrt {2\sqrt {2 \times 2} } } } } = \sqrt {2\sqrt {2\sqrt {2\sqrt {2 \times 2} } } } = \\\sqrt {2\sqrt {2\sqrt {2 \times 2} } } = \sqrt {2\sqrt {2 \times 2} } = \sqrt {2 \times 2} = 2\\\\2)\sqrt {\frac{{25}}{{16}}} + \sqrt {\frac{9}{{25}}} - \sqrt {\frac{{16}}{9}} = \frac{{\sqrt {25} }}{{\sqrt {16} }} + \frac{{\sqrt 9 }}{{\sqrt {25} }} - \frac{{\sqrt {16} }}{{\sqrt 9 }} = \\\frac{5}{4} + \frac{3}{5} - \frac{4}{3} = \frac{{75 + 36 - 80}}{{60}} = \frac{{31}}{{60}}\end{array}\)