Рационал алгебралық өрнектерді ықшамдау (Рустюмова 1.2.3 A (1-23))

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Өрнектерді ықшамдаңыз.

№ 1 Өрнекті ықшамдаңыз: ${\dfrac{{{m^4} - 49}}{{{m^2} + 7}} - \dfrac{{{m^6} - 343}}{{{m^4} + 7{m^2} + 49}}}$

Шешуі: $$ = \dfrac{{\left( {{m^2} - 7} \right)\left( {{m^2} + 7} \right)}}{{{m^2} + 7}} - \dfrac{{\left( {{m^2} - 7} \right)\left( {{m^4} + 7{m^2} + 49} \right)}}{{{m^4} + 7{m^2} + 49}}$$ $${ = {m^2} - 7 - \left( {{m^2} - 7} \right) = {m^2} - 7 - {m^2} + 7 = 0}$$

№ 2 Өрнекті ықшамдаңыз: ${\left( {\dfrac{{1 + n}}{{{n^2} - mn}} - \dfrac{{1 - m}}{{{m^2} - mn}}} \right) \cdot {{\left( {\dfrac{{m + n}}{{{m^2}n - {n^2}m}}} \right)}^{ - 1}}}$

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

№ 3 Өрнекті ықшамдаңыз: $\left( {\dfrac{{ab}}{{a - b}} + a} \right) \cdot \left( {\dfrac{{ab}}{{a + b}} - a} \right):\dfrac{{{a^2}{b^2}}}{{{b^2} - {a^2}}}$

Шешуі: $$ = \dfrac{{ab + {a^2} - ab}}{{a - b}} \cdot \dfrac{{ab - {a^2} - ab}}{{a + b}} \cdot \dfrac{{{b^2} - {a^2}}}{{a^2 {b^2}}} = $$ $${\dfrac{{{a^2}}}{{a - b}} \cdot \dfrac{{ - {a^2}}}{{a + b}} \cdot \dfrac{{{b^2} - {a^2}}}{{a^2 {b^2}}} = - \dfrac{{{a^4}}}{{{a^2} - {b^2}}} \cdot \dfrac{{{b^2} - {a^2}}}{{{a^2}{b^2}}} = }$$ $${ = \dfrac{{{a^4}}}{{{b^2} - {a^2}}} \cdot \dfrac{{{b^2} - {a^2}}}{{{a^2}{b^2}}} = \dfrac{{{a^2}}}{{{b^2}}} = {{\left( {\dfrac{a}{b}} \right)}^2}}$$

№ 4 Егер $b=-0,5$ болса, $\dfrac{{{{(a + b)}^2} - {{(ab + 1)}^2}}}{{{a^2} - 1}}:0,75$ есептеңіз.

Шешуі: $$ = \dfrac{{{a^2} + 2ab + {b^2} - {a^2}{b^2} - 2ab - 1}}{{{a^2} - 1}}:\dfrac{3}{4} = $$ $$ = \dfrac{{{a^2} + {b^2} - {a^2}{b^2} - 1}}{{{a^2} - 1}}:\dfrac{3}{4} = \dfrac{{{a^2} - {a^2}{b^2} + {b^2} - 1}}{{{a^2} - 1}} \cdot \dfrac{4}{3} = $$ $$ = \dfrac{{{a^2}\left( {1 - {b^2}} \right) - \left( {1 - {b^2}} \right)}}{{{a^2} - 1}} \cdot \dfrac{4}{3} = \dfrac{{\left( {1 - {b^2}} \right)\left( {{a^2} - 1} \right)}}{{{a^2} - 1}} \cdot \dfrac{4}{3} = $$ $$ = \left( {1 - {b^2}} \right) \cdot \dfrac{4}{3} = \left( {1 - {{\left( { - \dfrac{1}{2}} \right)}^2}} \right) \cdot \dfrac{4}{3} = \left( {1 - \dfrac{1}{4}} \right) \cdot \dfrac{4}{3} = \dfrac{3}{4} \cdot \dfrac{4}{3} = 1$$

№ 5 Өрнекті ықшамдаңыз: $\left( {\dfrac{{2ab}}{{{a^2} - {b^2}}} + \dfrac{{a - b}}{{2(a + b)}}} \right) \cdot \dfrac{{2a}}{{a + b}} - \dfrac{b}{{a - b}}$

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

№ 6 Егер $x=3$ болса,${\dfrac{{3x + 2}}{{2x + 3}} - \dfrac{{4x - 1}}{{2x + 3}} + \dfrac{{2{x^2} + 3x}}{{4{x^2} + 12x + 9}}}$ есептеңіз.

Шешуі: $$ = \dfrac{{3x + 2 - 4x + 1}}{{2x + 3}} + \dfrac{{x(2x + 3)}}{{{{(2x + 3)}^2}}} = $$ $$ = \dfrac{{3 - x}}{{2x + 3}} + \dfrac{x}{{2x + 3}} = \dfrac{3}{{2x + 3}}$$ $$x = 3\quad \Rightarrow \quad \dfrac{3}{{2 \cdot 3 + 3}} = \dfrac{3}{9} = \dfrac{1}{3}$$

№ 7 Өрнекті ықшамдаңыз: $\dfrac{6}{{7(x - 3)}} - \dfrac{1}{{{x^2} - 6x + 9}} + \dfrac{1}{{{x^2} - 9}}$

Шешуі: $$ = \dfrac{6}{{7(x - 3)}} - \dfrac{1}{{{{(x - 3)}^2}}} + \dfrac{1}{{(x - 3)(x + 3)}} = $$ $$ = \dfrac{{6{x^2} - 54 - 7x - 21 + 7x - 21}}{{7{{(x - 3)}^2}(x + 3)}} = \dfrac{{6{x^2} - 96}}{{7{{(x - 3)}^2}(x + 3)}} = $$ $$ = \dfrac{{6\left( {{x^2} - 16} \right)}}{{7{{(x - 3)}^2}(x + 3)}}$$ $$x = 4\quad \Rightarrow \quad = \dfrac{{6\left( {{4^2} - 16} \right)}}{{7{{(4 - 3)}^2}\left( {4 + 3} \right)}} = 0$$

№ 8 Өрнекті ықшамдаңыз: $\dfrac{{3a - 4}}{{a + 1}} + \dfrac{a}{{a + 1}}:\dfrac{a}{{{a^2} - 1}} + \dfrac{{5 - 2a}}{{a + 1}} = $

Шешуі: $$ = \dfrac{{3a - 4}}{{a + 1}} + \dfrac{a}{{a + 1}} \cdot \dfrac{{(a + 1)(a - 1)}}{a} + \dfrac{{5 - 2a}}{{a + 1}} = $$ $$ = \dfrac{{3a - 4 + 5 - 2a}}{{a + 1}} + \dfrac{{a - 1}}{1} = \dfrac{{a + 1}}{{a + 1}} + a - 1 = $$ $$ = 1 + a - 1 = a$$

№ 9 Өрнекті ықшамдаңыз: $\left( {\dfrac{1}{{{a^2}}} + \dfrac{1}{{{b^2}}} - \dfrac{1}{{ab}}} \right):\left( {\dfrac{{{a^2}}}{b} + \dfrac{{{b^2}}}{a}} \right)$

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

№ 10 Өрнекті ықшамдаңыз: $\left( {\dfrac{{{m^3} + {n^3}}}{{{m^2} - {n^2}}} - \dfrac{{{m^2} - {n^2}}}{{m + n}}} \right) \cdot {(mn)^{ - 1}}$

Шешуі: $$ = \left( {\dfrac{{(m + n)\left( {{m^2} - mn + {n^2}} \right)}}{{(m + n)(m - n)}} - \dfrac{{(m + n)(m - n)}}{{m + n}}} \right) \cdot \dfrac{1}{{mn}} = $$ $${ = \left( {\dfrac{{{m^2} - mn + {n^2}}}{{m - n}} - m + n} \right) \cdot \dfrac{1}{{mn}} = }$$ $$ = \dfrac{{{m^2} - mn + {n^2} - {{(m - n)}^2}}}{{m - n}} \cdot \dfrac{1}{{mn}} = $$ $$ = \dfrac{{{m^2} - mn + {n^2} - {m^2} + 2mn - {n^2}}}{{m - n}} \cdot \dfrac{1}{{mn}} = $$ $${ = \dfrac{{mn}}{{m - n}} \cdot \dfrac{1}{{mn}} = \dfrac{1}{{m - n}}}$$

№ 11 Өрнекті ықшамдаңыз: ${\left( {\dfrac{6}{{a - 3}} - \dfrac{3}{{a - 1}} - \dfrac{{a + 1}}{{{a^2} - 4a + 3}}} \right) \cdot {{\left( {\dfrac{{a + 1}}{{a - 3}}} \right)}^{ - 1}}}$

Шешуі: $${ = \left( {\dfrac{6}{{a - 1}} - \dfrac{3}{{a - 1}} - \dfrac{{a + 1}}{{(a - 3)(a - 1)}}} \right) \cdot \dfrac{{a - 3}}{{a + 1}} = }$$ $${ = \dfrac{{6a - 6 - 3a + 9 - a - 1}}{{(a - 3)(a - 1)}} \cdot \dfrac{{a - 3}}{{a + 1}} = \dfrac{{2a + 2}}{{(a - 3)(a - 1)}} \cdot \dfrac{{a - 3}}{{a + 1}} = }$$ $${ = \dfrac{{2(a + 1)}}{{a - 1}} \cdot \dfrac{1}{{(a + 1)}} = \dfrac{2}{{a - 1}}}$$

№ 12 Өрнекті ықшамдаңыз: ${\dfrac{{3{a^2} + 2ax - {x^2}}}{{(3x + a)(a + x)}} + 10 \cdot \dfrac{{ax - 3{x^2}}}{{{a^2} - 9{x^2}}} = }$

Шешуі: $${ = \dfrac{{ - (x - 3a)(x + a)}}{{(3x + a)(a + x)}} + 10 \cdot \dfrac{{x(a - 3x)}}{{(a - 3x)(a + 3x)}} = }$$ $${ = \dfrac{{3a - x}}{{3x + a}} + 10 \cdot \dfrac{x}{{3x + a}} = \dfrac{{3a - x + 10x}}{{3x + a}} = }$$ $${ = \dfrac{{9x + 3a}}{{3x + a}} = \dfrac{{3(3x + a)}}{{3x + a}} = 3}$$

№ 13 Өрнекті ықшамдаңыз: $\dfrac{{{{(a + b)}^2} - 3ab}}{{{a^3} + {a^2}b + a{b^2} + {b^3}}} \cdot \dfrac{{{a^4} - {b^4}}}{{{a^3} + {b^3}}} = $

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

№ 14 Өрнекті ықшамдаңыз: $\dfrac{{{m^2} - 9}}{{{m^2} - 1}}:\dfrac{{{m^2} + 4m + 3}}{{{m^2} - 4m + 3}}$

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

№ 15 Өрнекті ықшамдаңыз: $\dfrac{x}{{{x^2} + {y^2}}} - \dfrac{{y{{(x - y)}^2}}}{{{x^4} - {y^4}}}$

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

№ 16 Егер $a=2,71 \quad b=1,29$ болса, $\dfrac{a^4-b^4}{(a+b)^2-2ab}$ есептеңіз.

Шешуі: $$\dfrac{{{a^4} - {b^4}}}{{{{(a + b)}^2} - 2ab}} = \dfrac{{\left( {{a^2} + {b^2}} \right)\left( {{a^2} - {b^2}} \right)}}{{{a^2} + 2a \cdot b + {b^2} - 2ab}} = $$ $$ = \dfrac{{\left( {{a^2} + {b^2}} \right)\left( {{a^2} - {b^2}} \right)}}{{{a^2} + {b^2}}} = {a^2} - {b^2} = $$ $$ = (a - b)(a + b) = \left| {\left| {\begin{array}{*{20}{l}}{a = 2,71}\\{b = 1,29}\end{array}} \right|} \right| = $$ $$=(2,71 - 1,29)(2,71 + 1,29) = 4 \cdot 1,42 = 5,68$$

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

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

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

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

№ 19 Өрнекті ықшамдаңыз: $\left( {\dfrac{b}{{{a^2} - ab}} - \dfrac{1}{{a - b}}} \right):\left( {\dfrac{{a + b}}{{{a^2} - ab}} - \dfrac{b}{{ab - {b^2}}}} \right)$

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

№ 20 Өрнекті ықшамдаңыз: $\dfrac{x}{{x - y}} + \dfrac{{{x^2} + {y^2}}}{{{y^2} - {x^2}}} + \dfrac{x}{{x + y}}$

Шешуі: $$ = \dfrac{x}{{x - y}} - \dfrac{{{x^2} + {y^2}}}{{(x - y)(x + y)}} + \dfrac{{x}}{{x + y}} = $$ $$ = \dfrac{{{x^2} + xy - {x^2} - {y^2} + {x^2} - xy}}{{(x - y)(x + y)}} = \dfrac{{{x^2} - {y^2}}}{{{x^2} - {y^2}}} = 1$$

№ 21 Өрнекті ықшамдаңыз: $\left( {\dfrac{{m - 2}}{{m + 2}} - \dfrac{{m + 2}}{{m - 2}}} \right):\dfrac{{8m}}{{{m^2} - 4}}$

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

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

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

№ 23 Өрнекті ықшамдаңыз: $\left( {\dfrac{{ab}}{{{a^2} - {b^2}}} - \dfrac{b}{{2a - 2b}}} \right):\dfrac{{2b}}{{{a^2} - {b^2}}}$

Шешуі: $$ = \left( {\dfrac{{ab}}{{(a - b)(a + b)}} - \dfrac{b}{{2(a - b)}}} \right):\dfrac{{2b}}{{(a - b)(a + b)}} = $$ $$ = \dfrac{{2ab - ab - {b^2}}}{{2(a - b)(a + b)}} \cdot \dfrac{{(a - b)(a + b)}}{{2b}} = $$ $$ = \dfrac{{ab - {b^2}}}{{4b}} = \dfrac{{b(a - b)}}{{4b}} = \dfrac{{a - b}}{4}$$

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