Рационал алгебралық өрнектерді ықшамдау (Рустюмова 1.2.3 B (24-40))

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Өрнектерді ықшамдаңыз.

№ 24 Өрнекті ықшамдаңыз: ${\left( {\dfrac{1}{{2 - 6a}} + \dfrac{1}{{27{a^3} - 1}}:\dfrac{{1 + 3a}}{{1 + 3a + 9{a^2}}}} \right) \cdot \dfrac{{2 + 6a}}{a}}$

Шешуі: $${ = \left( {\dfrac{1}{{ - 2(3a - 1)}} + \dfrac{1}{{(3a - 1)\left( {9{a^2} + 3a + 1} \right)}} \cdot \dfrac{{1 + 3a + 9{a^2}}}{{3a + 1}}} \right) \cdot \dfrac{{2(3a + 1)}}{a} = }$$ $${ = \left( {\dfrac{1}{{ - 2(3a - 1)}} + \dfrac{1}{{(3a - 1)(3a + 1)}}} \right) \cdot \dfrac{{2(3a + 1)}}{a} = }$$ $${ = \dfrac{{3a + 1 - 2}}{{ - 2(3a - 1)(3a + 1)}} \cdot \dfrac{{2(3a + 1)}}{a} = \dfrac{{3a - 1}}{{ - 2(3a - 1)}} \cdot \dfrac{2}{a} = - \dfrac{1}{a}}$$

№ 25 Өрнекті ықшамдаңыз: $\dfrac{{8 - {n^3}}}{{2 + n}}:\left( {2 + \dfrac{{{n^2}}}{{2 + n}}} \right) - \dfrac{{{n^2}}}{{n - 2}} \cdot \dfrac{{4 - {n^2}}}{{{n^2} + 2n}}$

Шешуі: $$ = \dfrac{{ - \left( {{n^3} - 8} \right)}}{{n + 2}}:\dfrac{{4 + 2n + {n^2}}}{{n + 2}} - \dfrac{{{n^2}}}{{n - 2}} \cdot \dfrac{{(2 - n)\left( {2 + n} \right)}}{{n(n + 2)}} = $$ $$ = - \dfrac{{(n - 2)\left( {{n^2} + 2n + 4} \right)}}{{n + 2}} \cdot \dfrac{{n + 2}}{{{n^2} + 2n + 4}} + \dfrac{{{n^2}(n - 2)}}{{n(n - 2)}} = $$ $$ = - n + 2 + n = 2$$

№ 26 Өрнекті ықшамдаңыз: ${\left( {\dfrac{{12}}{{5{a^2} + a - 4}} - \dfrac{{a + 1}}{{3(5a - 4)}}} \right) \cdot \left( {\dfrac{{15a - 12}}{{a + 7}}} \right)}$

Шешуі: $${ = \left( {\dfrac{{12}}{{(a + 1)(5a - 4)}} - \dfrac{{a + 1}}{{3(5a - 4)}}} \right) \cdot \dfrac{{3(5a - 4)}}{{a + 7}} = }$$ $$ = \dfrac{{36 - {a^2} - 2a - 1}}{{3(a + 1)(5a - 4)}} \cdot \dfrac{{3(5a - 4)}}{{a + 7}} = \dfrac{{35 - 2a - {a^2}}}{{(a + 1)(a + 7)}} = $$ $$ = \dfrac{{ - (a + 7)(a - 5)}}{{(a + 1)(a + 7)}} = \dfrac{{5 - a}}{{a + 1}}$$

№ 27 Өрнекті ықшамдаңыз: ${\left( {\dfrac{{x + 5}}{{(x - 9)(x + 9)}} + \dfrac{{x + 7}}{{{{(x - 9)}^2}}}} \right) \cdot {{\left( {\dfrac{{x - 9}}{{x + 3}}} \right)}^2} + \dfrac{{7 + x}}{{9 + x}}}$

Шешуі: $${ = \dfrac{{{x^2} - 4x - 45 + {x^2} + 16x + 63}}{{(x + 9){{(x - 9)}^2}}} \cdot \dfrac{{{{(x - 9)}^2}}}{{{{(x + 3)}^2}}} + \dfrac{{7 + x}}{{9 + x}} = }$$ $${ = \dfrac{{2{x^2} + 12x + 18}}{{x + 9}} \cdot \dfrac{1}{{{{(x + 3)}^2}}} + \dfrac{{7 + x}}{{9 + x}} = }$$ $${ = \dfrac{{2\left( {{x^2} + 6x + 9} \right)}}{{x + 9}} \cdot \dfrac{1}{{{{(x + 3)}^2}}} + \dfrac{{7 + x}}{{9 + x}} = \dfrac{{2{{(x + 3)}^2}}}{{x + 9}} \cdot \dfrac{1}{{{{(x + 3)}^2}}} + \dfrac{{7 + x}}{{9 + x}} = }$$ $${ = \dfrac{2}{{x + 9}} + \dfrac{{7 + x}}{{9 + x}} = \dfrac{{2 + 7 + x}}{{x + 9}} = \dfrac{{x + 9}}{{x + 9}} = 1}$$

№ 28 $a=-2,5; \quad b=0,5$ болғанда $\left( {\dfrac{{{a^2}}}{{a + b}} - \dfrac{{{a^3}}}{{{a^2} + 2ab + {b^2}}}} \right):\left( {\dfrac{a}{{a + b}} - \dfrac{{{a^2}}}{{{a^2} - {b^2}}}} \right)$ есептеңіз.

Шешуі: $$ = \left( {\dfrac{{{a^2}}}{{a + b}} - \dfrac{{{a^3}}}{{{{(a + b)}^2}}}} \right):\left( {\dfrac{a}{{a + b}} - \dfrac{{{a^2}}}{{(a + b)(a - b)}}} \right) = $$ $$ = \dfrac{{{a^3} + {a^2}b - {a^3}}}{{{{(a + b)}^2}}}:\dfrac{{{a^2} - ab - {a^2}}}{{(a + b)(a - b)}} = $$ $$ = \dfrac{{{a^2}b}}{{{{(a + b)}^2}}} \cdot \dfrac{{(a + b)(a - b)}}{{ - ab}} = \dfrac{{a(a - b)}}{{ - (a + b)}} = \dfrac{{a(b - a)}}{{a + b}} = $$ $$ = \left\| {\begin{array}{*{20}{l}}{a = - 2,5}\\{b = 0,5}\end{array}} \right\| = \dfrac{{ - 2,5(0,5 + 2,5)}}{{ - 2,5 + 0,5}} = \dfrac{{ - 2,5 \cdot 3}}{{ - 2}} = 3,75$$

№ 29 Өрнекті ықшамдаңыз: ${\left( {\dfrac{{49}}{{{a^3} + 27}} - \dfrac{{a + 3}}{{{a^2} + 9 - 3a}}} \right) \cdot \dfrac{{a\left( {{a^3} + 27} \right)}}{ {16 - {a^2}}} + \dfrac{{4 - 9a - {a^2}}}{{a + 4}}}$

Шешуі: $${ = \left( {\dfrac{{49}}{{(a + 3)\left( {{a^2} - 3a + 9} \right)}} - \dfrac{{a + 3}}{{{a^2} - 3a + 9}}} \right) \cdot \dfrac{{a\left( {{a^3} + 27} \right)}}{{16 - {a^2}}} + \dfrac{{4 - 9a - {a^2}}}{{a + 4}} = }$$ $${ = \dfrac{{49 - {{(a + 3)}^2}}}{{(a + 3)\left( {{a^2} - 3a + 9} \right)}} \cdot \dfrac{{a(a + 3)\left( {{a^2} - 3a + 9} \right)}}{{(4 - a)(4 + a)}} + \dfrac{{4 - 9a - {a^2}}}{{a + 4}} = }$$ $${ = \dfrac{{(7 - a - 3)(7 + a + 3) \cdot a}}{{(4 - a)(a + 4)}} + \dfrac{{4 - 9a - {a^2}}}{{a + 4}} = }$$ $${ = \dfrac{{(4 - a)(10 + a) \cdot a}}{{(4 - a)(a + 4)}} + \dfrac{{4 - 9a - {a^2}}}{{a + 4}} = }$$ $${ = \dfrac{{a(10 + a)}}{{a + 4}} + \dfrac{{4 - 9a - {a^2}}}{{a + 4}} = \dfrac{{10a + {a^2} + 4 - 9a - {a^2}}}{{a + 4}} = \dfrac{{a + 4}}{{a + 4}} = 1}$$

№ 30 Өрнекті ықшамдаңыз: $(x + 1) \cdot \left( {\dfrac{1}{{x + 1}} + \dfrac{4}{{{x^2} - 4x}} - \dfrac{5}{{{x^2} - 3x - 4}}} \right):\left( {1 - \dfrac{1}{x}} \right)$

Шешуі: $$ = (x + 1) \cdot \left( {\dfrac{1}{{x + 1}} + \dfrac{4}{{x(x - 4)}} - \dfrac{5}{{(x + 1)(x - 4)}}} \right):\dfrac{{x - 1}}{x} = $$ $$ = (x + 1) \cdot \dfrac{{x(x - 4) + 4(x + 1) - 5x}}{{x(x + 1)(x - 4)}} \cdot \dfrac{x}{{x - 1}} = $$ $$ = \dfrac{{x + 1}}{1} \cdot \dfrac{{{x^2} - 4x + 4x + 4 - 5x}}{{x(x + 1)(x - 4)}} \cdot \dfrac{x}{{x - 1}} = $$ $$ = \dfrac{{{x^2} - 5x + 4}}{{x(x - 4)}} \cdot \dfrac{x}{{x - 1}} = \dfrac{{(x - 1)(x - 4)}}{{x(x - 4)}} \cdot \dfrac{x}{{x - 1}} = 1$$

№ 31 Өрнекті ықшамдаңыз: ${\left( {\dfrac{2}{a} - \dfrac{a}{4} - \dfrac{3}{{2a}} + \dfrac{1}{{2a}}} \right):\dfrac{{2 - a}}{4} - \left( {1 + \dfrac{2}{a}} \right)}$

Шешуі: $${ = \dfrac{{8 - {a^2} - 6 + 2}}{{4a}} \cdot \dfrac{4}{{2 - a}} - \dfrac{{a + 2}}{a} = }$$ $$ = \dfrac{{4 - {a^2}}}{{4a}} \cdot \dfrac{4}{{2 - a}} - \dfrac{{a + 2}}{a} = $$ $$ = \dfrac{{(2 - a)(2 + a)}}{{4a}} \cdot \dfrac{4}{{2 - a}} - \dfrac{{a + 2}}{a} = \dfrac{{a + 2}}{a} - \dfrac{{a + 2}}{a} = 0$$

№ 32 Өрнекті ықшамдаңыз: ${\left( {\dfrac{8}{{2{a^2} - 8a}} - \dfrac{{3a + 32}}{{{a^3} - 64}}} \right):\dfrac{{a - 8}}{{{a^3} + 4{a^2} + 16a}} - \dfrac{4}{{4 - a}}}$

Шешуі: $${ = \left( {\dfrac{8}{{2a(a - 4)}} - \dfrac{{3a + 32}}{{(a - 4)\left( {{a^2} + 4a + 16} \right)}}} \right) \cdot \dfrac{{{a^3} + 4{a^2} + 16a}}{{a - 8}} + \dfrac{4}{{a - 4}} = }$$ $${ = \left( {\dfrac{4}{{a(a - 4)}} - \dfrac{{3a + 32}}{{(a - 4)({a^2} + 4a + 16)}}} \right) \cdot \dfrac{{a\left( {{a^2} + 4a + 16} \right)}}{{a - 8}} + \dfrac{4}{{a - 4}} = }$$ $${ = \dfrac{{4\left( {{a^2} + 4a + 16} \right) - a(3a + 32)}}{{a(a - 4)\left( {{a^2} + 4a + 16} \right)}} \cdot \dfrac{{a\left( {{a^2} + 4a + 16} \right)}}{{a - 8}} + \dfrac{4}{a-4}=}$$ $${ = \dfrac{{4{a^2} + 16a + 64 - 3{a^2} - 32a}}{{a - 4}} \cdot \dfrac{1}{{a - 8}} + \dfrac{4}{{a - 4}} = }$$ $${ = \dfrac{{{a^2} - 16a + 64}}{{a - 4}} \cdot \dfrac{1}{{a - 8}} + \dfrac{4}{{a - 4}} = \dfrac{{{{(a - 8)}^2}}}{{a - 4}} \cdot \dfrac{1}{{a - 8}} + \dfrac{4}{{a - 4}} = }$$ $${ = \dfrac{{a - 8}}{{a - 4}} + \dfrac{4}{{a - 4}} = \dfrac{{a - 8 + 4}}{{a - 4}} = \dfrac{{a - 4}}{{a - 4}} = 1}$$

№ 33 Өрнекті ықшамдаңыз: $\dfrac{{4{a^2} - 1}}{{{a^2} - 1}}:\left( {\dfrac{a}{{{a^2} - 2a + 1}} + \dfrac{a}{{{a^2} - 1}} - \dfrac{2}{{a + 1}}} \right)$

Шешуі: $$ = \dfrac{{(2a - 1)(2a + 1)}}{{(a - 1)(a + 1)}}:\left( {\dfrac{a}{{{{(a - 1)}^2}}} + \dfrac{a}{{(a - 1)(a + 1)}} - \dfrac{2}{{a + 1}}} \right) = $$ $$ = \dfrac{{(2a - 1)(2a + 1)}}{{(a - 1)(a + 1)}}:\dfrac{{a(a + 1) + a(a - 1) - 2{{(a - 1)}^2}}}{{{{(a - 1)}^2}(a + 1)}} = $$ $$ = \dfrac{{(2a - 1)(2a + 1)}}{{(a - 1)(a + 1)}}:\dfrac{{{a^2} + a + {a^2} - a - 2{{(a - 1)}^2}}}{{{{(a - 1)}^2}(a + 1)}} = $$ $$ = \dfrac{{(2a - 1)(2a + 1)}}{{(a - 1)(a + 1)}} \cdot \dfrac{{{{(a - 1)}^2}(a + 1)}}{{2{a^2} - 2{{(a - 1)}^2}}} = $$ $$ = (2a - 1)(2a + 1) \cdot \dfrac{{a - 1}}{{2\left( {{a^2} - {{(a - 1)}^2}} \right)}} = $$ $$ = (2a - 1)(2a + 1) \cdot \dfrac{{a - 1}}{{2(a - a + 1)(a + a - 1)}} = $$ $$ = \left( {2a - 1} \right)\left( {2a + 1} \right) \cdot \frac{{a - 1}}{{2\left( {2a - 1} \right)}} = \left( {2a + 1} \right) \cdot \frac{{a - 1}}{2} = $$ $$ = \frac{{\left( {2a + 1} \right)\left( {a - 1} \right)}}{2} = \frac{{2{a^2} - 2a + a - 1}}{2} = \frac{{2{a^2} - a - 1}}{2}$$

№ 34 Өрнекті ықшамдаңыз: $\dfrac{{{{\left( {{x^2} + 5} \right)}^2} + 4\left( {{x^2} + 5} \right) + 4}}{{{x^2} - 10x + 21}} \cdot \dfrac{{2{x^2} - 18}}{{{{\left( {{x^2} + 7} \right)}^2}}}$

Шешуі: $$ = \dfrac{{{x^4} + 10{x^2} + 25 + 4{x^2} + 20 + 4}}{{(x - 3)(x - 7)}} \cdot \dfrac{{2\left( {{x^2} - 9} \right)}}{{{{\left( {{x^2} + 7} \right)}^2}}} = $$ $$ = \dfrac{{{x^4} + 14{x^2} + 49}}{{(x - 3)(x - 7)}} \cdot \dfrac{{2(x - 3)(x + 3)}}{{{{\left( {{x^2} + 7} \right)}^2}}} = $$ $$ = \dfrac{{{{\left( {{x^2} + 7} \right)}^2}}}{{(x - 3)(x - 7)}} \cdot \dfrac{{2(x - 3)(x + 3)}}{{{{\left( {{x^2} + 7} \right)}^2}}} = \dfrac{{2(x + 3)}}{{x - 7}} = \dfrac{{2x + 6}}{{x - 7}}$$

№ 35 Өрнекті ықшамдаңыз: $\dfrac{{{y^2} + 18y + 77}}{{{y^2} - 49}} \cdot \dfrac{{5y - 35}}{{{{(y + 8)}^2} + 6(y + 8) + 9}}$

Шешуі: $$ = \dfrac{{{y^2} + 18y + 77}}{{(y - 7)(y + 7)}} \cdot \dfrac{{5(y - 7)}}{{((y + 8) + 3)^2}} = \dfrac{{{y^2} + 18y + 77}}{{y + 7}} \cdot \dfrac{5}{{{{(y + 11)}^2}}} = $$ $$ = \dfrac{{(y + 7)(y + 11)}}{{y + 7}} \cdot \dfrac{5}{{{{(y + 11)}^2}}} = \dfrac{5}{{y + 11}}$$

№ 36 Өрнекті ықшамдаңыз: $\left( {\dfrac{1}{{{{(2 - a)}^2}}} - \dfrac{2}{{{a^2} - 4}} + \dfrac{1}{{{{(2 + a)}^2}}}} \right) \cdot {\left( {{a^2} - 4} \right)^2}$

Шешуі: $$ = \left( {\dfrac{1}{{{{(a - 2)}^2}}} - \dfrac{2}{{(a - 2)(a + 2)}} + \dfrac{1}{{{{(a + 2)}^2}}}} \right) \cdot \left((a - 2)(a + 2)\right)^2 = $$ $$ = \dfrac{{{{(a + 2)}^2} - 2(a - 2)(a + 2) + {{(a - 2)}^2}}}{{{{(a - 2)}^2}{{(a + 2)}^2}}} \cdot {(a - 2)^2} \cdot {(a + 2)^2} = $$ $$ = {a^2} + 4a + 4 - 2{a^2} + 8 + {a^2} - 4a + 4 = 16$$

№ 37 Өрнекті ықшамдаңыз: ${\dfrac{1}{{(a - b)(a - c)}} + \dfrac{1}{{(b - c)(b - a)}} + \dfrac{1}{{(c - a)(c - b)}}}$

Шешуі: $${ = \dfrac{1}{{(a - b)(a - c)}} - \dfrac{1}{{(a - b)(b - c)}} + \dfrac{1}{{(a - c)(b - c)}} = }$$ $${ = \dfrac{{(b - c) - (a - c) + (a - b)}}{{(a - b)(a - c)(b - c)}} = \dfrac{{b - c - a + c + a - b}}{{(a - b)(a - c)(b - c)}} = }$$ $${ = \dfrac{0}{{(a - b)(a - c)(b - c)}} = 0}$$

№ 38 Өрнекті ықшамдаңыз: $\left( {{x^2} - {y^2} - {z^2} + 2yz} \right):\dfrac{{x + y - z}}{{x + y + z}}$

Шешуі: $$ = \left( {{x^2} - \left( {{y^2} - 2yz + {z^2}} \right)} \right):\dfrac{{x + y - z}}{{x + y + z}} = $$ $${ = \left( {{x^2} - {{(y - z)}^2}} \right) \cdot \dfrac{{x + y + z}}{{x + y - z}} = }$$ $$ = (x - y + z)(x + y - z) \cdot \dfrac{{x + y + z}}{{x + y - z}} = $$ $$ = ((x + z) - y) \cdot ((x + z) + y) = {(x + z)^2} - {y^2}$$

№ 39 Өрнекті ықшамдаңыз: ${\left( {\dfrac{{2x + 1}}{{x + 2}} - \dfrac{{4x + 2}}{{4 - {x^2}}}} \right):\dfrac{{2x + 1}}{{x - 2}} + \dfrac{2}{{x + 2}}}$

Шешуі: $${ = \left( {\dfrac{{2x + 1}}{{x + 2}} + \dfrac{{4x + 2}}{{{x^2} - 4}}} \right) \cdot \dfrac{{x - 2}}{{2x + 1}} + \dfrac{2}{{x + 2}} = }$$ $${ = \left( {\dfrac{{2x + 1}}{{x + 2}} + \dfrac{{4x + 2}}{{(x - 2)(x + 2)}}} \right) \cdot \dfrac{{x - 2}}{{2x + 1}} + \dfrac{2}{{x + 2}} = }$$ $${ = \dfrac{{(2x + 1)(x - 2) + 2(2x + 1)}}{{(x - 2)(x + 2)}} \cdot \dfrac{{x - 2}}{{2x + 1}} + \dfrac{2}{{x + 2}} = }$$ $$ = \dfrac{{(2x + 1)(x - 2 + 2)}}{{x + 2}} \cdot \dfrac{1}{{2x + 1}} + \dfrac{2}{{x + 2}} = $$ $$ = \dfrac{x}{{x + 2}} + \dfrac{2}{{x + 2}} = \dfrac{{x + 2}}{{x + 2}} = 1$$

№ 40 Өрнекті ықшамдаңыз: $\left( {\dfrac{9}{{{x^2} - 9}} + \dfrac{3}{{{{(3 - x)}^2}}}} \right):\dfrac{6}{{{{(x - 3)}^2}}} + \dfrac{{1 - 2x}}{{3 + x}}$

Шешуі: $$ = \left( {\dfrac{9}{{(x - 3)(x + 3)}} + \dfrac{3}{{{{(x - 3)}^2}}}} \right) \cdot \dfrac{{{{(x - 3)}^2}}}{6} + \dfrac{{1 - 2x}}{{3 + x}} = $$ $$ = \dfrac{{9(x - 3) + 3(x + 3)}}{{{{(x - 3)}^2}(x + 3)}} \cdot \dfrac{{{{(x - 3)}^2}}}{6} + \dfrac{{1 - 2x}}{{3 + x}} = $$ $$ = \dfrac{{9x - 27 + 3x + 9}}{{x + 3}} \cdot \dfrac{1}{6} + \dfrac{{1 - 2x}}{{3 + x}} = $$ $$ = \dfrac{{12x - 18}}{{x + 3}} \cdot \dfrac{1}{6} + \dfrac{{1 - 2x}}{{3 + x}} = \dfrac{{6(2x - 3)}}{{x + 3}} \cdot \dfrac{1}{6} + \dfrac{{1 - 2x}}{{3 + x}} = $$ $$ = \dfrac{{2x - 3}}{{x + 3}} + \dfrac{{1 - 2x}}{{x + 3}} = \dfrac{{2x - 3 + 1 - 2x}}{{x + 3}} = - \dfrac{2}{{x + 3}}$$

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