1.Упростите выражение :
((x/y) + (y/x))умножить на (x^2y^2/x^2+y^2) - xy
2.Упростите выражение:
((2ab/a^3-b^3)+(a-b/a^2+ab+b^2)):a^2+b^2/a-b
3.Докажите тождество:
((1/x)-(1/x^2)):((x-1)/x)-(1/x)+(x/x-2)-(x/x+2)=4x/x^2-4
Ответы на вопрос
Ответил АннаАрт
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1.Упростите выражение :
![(frac{x}{y}+frac{y}{x})(frac{x^2y^2}{x^2+y^2})-xy \ \
1) frac{x}{y}+frac{y}{x}= frac{x^2+y^2}{xy} \ \
2) frac{x^2+y^2}{xy}*frac{x^2y^2}{x^2+y^2}=xy \ \
3) xy-xy=0](https://tex.z-dn.net/?f=%28frac%7Bx%7D%7By%7D%2Bfrac%7By%7D%7Bx%7D%29%28frac%7Bx%5E2y%5E2%7D%7Bx%5E2%2By%5E2%7D%29-xy+%5C++%5C+%0A1%29+frac%7Bx%7D%7By%7D%2Bfrac%7By%7D%7Bx%7D%3D+frac%7Bx%5E2%2By%5E2%7D%7Bxy%7D++%5C++%5C+%0A2%29+frac%7Bx%5E2%2By%5E2%7D%7Bxy%7D%2Afrac%7Bx%5E2y%5E2%7D%7Bx%5E2%2By%5E2%7D%3Dxy+%5C++%5C+%0A3%29+xy-xy%3D0 "(frac{x}{y}+frac{y}{x})(frac{x^2y^2}{x^2+y^2})-xy \ \
1) frac{x}{y}+frac{y}{x}= frac{x^2+y^2}{xy} \ \
2) frac{x^2+y^2}{xy}*frac{x^2y^2}{x^2+y^2}=xy \ \
3) xy-xy=0")
2.Упростите выражение:
![(frac{2ab}{a^3-b^3}+frac{a-b}{a^2+ab+b^2}):frac{a^2+b^2}{a-b} \ \
1) frac{2ab}{a^3-b^3}+frac{a-b}{a^2+ab+b^2}= frac{2ab}{(a-b)(a^2+ab+b^2)}+frac{a-b}{a^2+ab+b^2}= \ \ =frac{2ab+(a-b)^2}{(a-b)(a^2+ab+b^2)}=frac{2ab+a^2-2ab+b^2}{(a-b)(a^2+ab+b^2)}=frac{a^2+b^2}{(a-b)(a^2+ab+b^2)} \ \
2) frac{a^2+b^2}{(a-b)(a^2+ab+b^2)}:frac{a^2+b^2}{a-b}=frac{a^2+b^2}{(a-b)(a^2+ab+b^2)}*frac{a-b}{a^2+b^2}= \ \
=frac{1}{a^2+ab+b^2}](https://tex.z-dn.net/?f=%28frac%7B2ab%7D%7Ba%5E3-b%5E3%7D%2Bfrac%7Ba-b%7D%7Ba%5E2%2Bab%2Bb%5E2%7D%29%3Afrac%7Ba%5E2%2Bb%5E2%7D%7Ba-b%7D+%5C++%5C+%0A1%29+frac%7B2ab%7D%7Ba%5E3-b%5E3%7D%2Bfrac%7Ba-b%7D%7Ba%5E2%2Bab%2Bb%5E2%7D%3D+frac%7B2ab%7D%7B%28a-b%29%28a%5E2%2Bab%2Bb%5E2%29%7D%2Bfrac%7Ba-b%7D%7Ba%5E2%2Bab%2Bb%5E2%7D%3D+%5C++%5C+%3Dfrac%7B2ab%2B%28a-b%29%5E2%7D%7B%28a-b%29%28a%5E2%2Bab%2Bb%5E2%29%7D%3Dfrac%7B2ab%2Ba%5E2-2ab%2Bb%5E2%7D%7B%28a-b%29%28a%5E2%2Bab%2Bb%5E2%29%7D%3Dfrac%7Ba%5E2%2Bb%5E2%7D%7B%28a-b%29%28a%5E2%2Bab%2Bb%5E2%29%7D+%5C++%5C+%0A2%29+frac%7Ba%5E2%2Bb%5E2%7D%7B%28a-b%29%28a%5E2%2Bab%2Bb%5E2%29%7D%3Afrac%7Ba%5E2%2Bb%5E2%7D%7Ba-b%7D%3Dfrac%7Ba%5E2%2Bb%5E2%7D%7B%28a-b%29%28a%5E2%2Bab%2Bb%5E2%29%7D%2Afrac%7Ba-b%7D%7Ba%5E2%2Bb%5E2%7D%3D+%5C++%5C+%0A%3Dfrac%7B1%7D%7Ba%5E2%2Bab%2Bb%5E2%7D "(frac{2ab}{a^3-b^3}+frac{a-b}{a^2+ab+b^2}):frac{a^2+b^2}{a-b} \ \
1) frac{2ab}{a^3-b^3}+frac{a-b}{a^2+ab+b^2}= frac{2ab}{(a-b)(a^2+ab+b^2)}+frac{a-b}{a^2+ab+b^2}= \ \ =frac{2ab+(a-b)^2}{(a-b)(a^2+ab+b^2)}=frac{2ab+a^2-2ab+b^2}{(a-b)(a^2+ab+b^2)}=frac{a^2+b^2}{(a-b)(a^2+ab+b^2)} \ \
2) frac{a^2+b^2}{(a-b)(a^2+ab+b^2)}:frac{a^2+b^2}{a-b}=frac{a^2+b^2}{(a-b)(a^2+ab+b^2)}*frac{a-b}{a^2+b^2}= \ \
=frac{1}{a^2+ab+b^2}")
3.Докажите тождество:
![(frac{1}{x}-frac{1}{x^2}):frac{x-1}{x}-frac{1}{x}+frac{x}{x-2}-frac{x}{x+2}=frac{4x}{x^2-4} \ \
1) frac{1}{x}-frac{1}{x^2}= frac{x-1}{x^2} \ \
2) frac{x-1}{x^2}:frac{x-1}{x}=frac{x-1}{x^2}*frac{x}{x-1}= frac{1}{x} \ \
3) frac{1}{x}-frac{1}{x}+frac{x}{x-2}-frac{x}{x+2}=frac{x}{x-2}-frac{x}{x+2}= frac{x(x+2)-x(x-2)}{(x-2)(x+2)} =frac{4x}{x^2-4}](https://tex.z-dn.net/?f=%28frac%7B1%7D%7Bx%7D-frac%7B1%7D%7Bx%5E2%7D%29%3Afrac%7Bx-1%7D%7Bx%7D-frac%7B1%7D%7Bx%7D%2Bfrac%7Bx%7D%7Bx-2%7D-frac%7Bx%7D%7Bx%2B2%7D%3Dfrac%7B4x%7D%7Bx%5E2-4%7D+%5C++%5C+%0A1%29+frac%7B1%7D%7Bx%7D-frac%7B1%7D%7Bx%5E2%7D%3D+frac%7Bx-1%7D%7Bx%5E2%7D++%5C++%5C+%0A2%29+frac%7Bx-1%7D%7Bx%5E2%7D%3Afrac%7Bx-1%7D%7Bx%7D%3Dfrac%7Bx-1%7D%7Bx%5E2%7D%2Afrac%7Bx%7D%7Bx-1%7D%3D+frac%7B1%7D%7Bx%7D++%5C++%5C+%0A3%29+frac%7B1%7D%7Bx%7D-frac%7B1%7D%7Bx%7D%2Bfrac%7Bx%7D%7Bx-2%7D-frac%7Bx%7D%7Bx%2B2%7D%3Dfrac%7Bx%7D%7Bx-2%7D-frac%7Bx%7D%7Bx%2B2%7D%3D+frac%7Bx%28x%2B2%29-x%28x-2%29%7D%7B%28x-2%29%28x%2B2%29%7D+%3Dfrac%7B4x%7D%7Bx%5E2-4%7D "(frac{1}{x}-frac{1}{x^2}):frac{x-1}{x}-frac{1}{x}+frac{x}{x-2}-frac{x}{x+2}=frac{4x}{x^2-4} \ \
1) frac{1}{x}-frac{1}{x^2}= frac{x-1}{x^2} \ \
2) frac{x-1}{x^2}:frac{x-1}{x}=frac{x-1}{x^2}*frac{x}{x-1}= frac{1}{x} \ \
3) frac{1}{x}-frac{1}{x}+frac{x}{x-2}-frac{x}{x+2}=frac{x}{x-2}-frac{x}{x+2}= frac{x(x+2)-x(x-2)}{(x-2)(x+2)} =frac{4x}{x^2-4}")
2.Упростите выражение:
3.Докажите тождество:
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