![\Big(\dfrac{8}{\sqrt{x}-1}+\dfrac{\sqrt[4]{x}+1}{\sqrt[4]{x}-1}-\dfrac{\sqrt[4]{x}+3}{\sqrt[4]{x}+1}\Big):\dfrac{3}{\sqrt{x}-1}=\\ \\ \\ =\Big(\dfrac{8}{\sqrt{x}-1}+\dfrac{(\sqrt[4]{x}+1)^2-(\sqrt[4]{x}+3)(\sqrt[4]{x}-1)}{(\sqrt[4]{x}-1)\cdot (\sqrt[4]{x}+1)}\Big)\cdot \dfrac{\sqrt{x}-1}{3}=\\ \\ \\=\Big(\dfrac{8}{\sqrt{x}-1}+\dfrac{\sqrt{x}+2\sqrt[4]{x}+1-(\sqrt{x}-\sqrt[4]{x}+3\sqrt[4]{x}-3)}{\sqrt{x}-1}\Big)\cdot \dfrac{\sqrt{x}-1}{3}=](https://tex.z-dn.net/?f=%5CBig%28%5Cdfrac%7B8%7D%7B%5Csqrt%7Bx%7D-1%7D%2B%5Cdfrac%7B%5Csqrt%5B4%5D%7Bx%7D%2B1%7D%7B%5Csqrt%5B4%5D%7Bx%7D-1%7D-%5Cdfrac%7B%5Csqrt%5B4%5D%7Bx%7D%2B3%7D%7B%5Csqrt%5B4%5D%7Bx%7D%2B1%7D%5CBig%29%3A%5Cdfrac%7B3%7D%7B%5Csqrt%7Bx%7D-1%7D%3D%5C%5C+%5C%5C+%5C%5C+%3D%5CBig%28%5Cdfrac%7B8%7D%7B%5Csqrt%7Bx%7D-1%7D%2B%5Cdfrac%7B%28%5Csqrt%5B4%5D%7Bx%7D%2B1%29%5E2-%28%5Csqrt%5B4%5D%7Bx%7D%2B3%29%28%5Csqrt%5B4%5D%7Bx%7D-1%29%7D%7B%28%5Csqrt%5B4%5D%7Bx%7D-1%29%5Ccdot+%28%5Csqrt%5B4%5D%7Bx%7D%2B1%29%7D%5CBig%29%5Ccdot+%5Cdfrac%7B%5Csqrt%7Bx%7D-1%7D%7B3%7D%3D%5C%5C+%5C%5C+%5C%5C%3D%5CBig%28%5Cdfrac%7B8%7D%7B%5Csqrt%7Bx%7D-1%7D%2B%5Cdfrac%7B%5Csqrt%7Bx%7D%2B2%5Csqrt%5B4%5D%7Bx%7D%2B1-%28%5Csqrt%7Bx%7D-%5Csqrt%5B4%5D%7Bx%7D%2B3%5Csqrt%5B4%5D%7Bx%7D-3%29%7D%7B%5Csqrt%7Bx%7D-1%7D%5CBig%29%5Ccdot+%5Cdfrac%7B%5Csqrt%7Bx%7D-1%7D%7B3%7D%3D)
![=\Big(\dfrac{8}{\sqrt{x}-1}+\dfrac{\sqrt{x}+2\sqrt[4]{x}+1-\sqrt{x}+\sqrt[4]{x}-3\sqrt[4]{x}+3}{\sqrt{x}-1}\Big)\cdot \dfrac{\sqrt{x}-1}{3}=\\ \\ \\ =\Big(\dfrac{8}{\sqrt{x}-1}+\dfrac{4}{\sqrt{x}-1}\Big)\cdot \dfrac{\sqrt{x}-1}{3}=\\ \\ \\ =\dfrac{12}{\sqrt{x}-1}\cdot \dfrac{\sqrt{x}-1}{3}=4](https://tex.z-dn.net/?f=%3D%5CBig%28%5Cdfrac%7B8%7D%7B%5Csqrt%7Bx%7D-1%7D%2B%5Cdfrac%7B%5Csqrt%7Bx%7D%2B2%5Csqrt%5B4%5D%7Bx%7D%2B1-%5Csqrt%7Bx%7D%2B%5Csqrt%5B4%5D%7Bx%7D-3%5Csqrt%5B4%5D%7Bx%7D%2B3%7D%7B%5Csqrt%7Bx%7D-1%7D%5CBig%29%5Ccdot+%5Cdfrac%7B%5Csqrt%7Bx%7D-1%7D%7B3%7D%3D%5C%5C+%5C%5C+%5C%5C+%3D%5CBig%28%5Cdfrac%7B8%7D%7B%5Csqrt%7Bx%7D-1%7D%2B%5Cdfrac%7B4%7D%7B%5Csqrt%7Bx%7D-1%7D%5CBig%29%5Ccdot+%5Cdfrac%7B%5Csqrt%7Bx%7D-1%7D%7B3%7D%3D%5C%5C+%5C%5C+%5C%5C+%3D%5Cdfrac%7B12%7D%7B%5Csqrt%7Bx%7D-1%7D%5Ccdot+%5Cdfrac%7B%5Csqrt%7Bx%7D-1%7D%7B3%7D%3D4)
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Использованы формулы
![\sqrt[4]{x}^2=\sqrt{x}\\ \\(a+b)^2=a^2+2ab+b^2 \\ \\(a-b)(a+b)=a^2-b^2](https://tex.z-dn.net/?f=%5Csqrt%5B4%5D%7Bx%7D%5E2%3D%5Csqrt%7Bx%7D%5C%5C+%5C%5C%28a%2Bb%29%5E2%3Da%5E2%2B2ab%2Bb%5E2+%5C%5C+%5C%5C%28a-b%29%28a%2Bb%29%3Da%5E2-b%5E2)
4с( во 2 степени)- 8с-с (во второй степени) +8с-16=3с(во 2 степени)-16
Воспользуемся так называемым методом спуска.
9x-11y=8
9x=11y+8
x=y+(2y+8)/9
2y+8 должно делиться на 9, поэтому:
2y+8=9k
2y=9k-8
y=4k-4+k/2
k должно делиться на 2, поэтому:
k=2n.
Спуск закончен.
y=8n-4+n=9n-4
x=11n-4.
Очевидно, что x и y - целые и неотрицательные при n>0. Поэтому в общем виде решение выглядит так:
(11n-4;9n-4), n>0.
Если нужно подобрать частные случаи, имеем:
(7;5), (18;14), (29;23) и т.д.
X^5-x^7+3x+С
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Полное реш. внутри
Надеюсь, помог
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