Дано:
рябин- 37
сосен- 17
берез- 4 часть от саженцев сосен
найти: сколько деревьев всего
решение:
1) 17*4=68- берез
2) 37+17+68= 122 дерева всего
ответ: 122 дерева посадили в парке.
90-40
Х-80
25+30
17+Х
Если нужно решить, напиши в км.
Так как под корнем чётной степени стоит отрицательное число, то вычисления будут в области комплексных чисел.
![\sqrt[4]{-625}=\sqrt[4]{-1\cdot625}=\sqrt[4]{-1\cdot5^4}=5\sqrt[4]{-1}](https://tex.z-dn.net/?f=%5Csqrt%5B4%5D%7B-625%7D%3D%5Csqrt%5B4%5D%7B-1%5Ccdot625%7D%3D%5Csqrt%5B4%5D%7B-1%5Ccdot5%5E4%7D%3D5%5Csqrt%5B4%5D%7B-1%7D)
Ниже приведены два способа извлечения корня из (-1).
В области комплексных чисел (-1) можно представить в алгебраической форме : -1 = -1 + 0·<em>i</em> , <u>a= -1</u>; <u>b = 0</u>
![\sqrt{-1}=\pm\Bigg(\sqrt{\dfrac{a+\sqrt{a^2+b^2}}2}+\sqrt{\dfrac{-a+\sqrt{a^2+b^2}}2}i\Bigg)=\\\\~~~=\pm\Bigg(\sqrt{\dfrac{-1+\sqrt{1+0}}2}+\sqrt{\dfrac{1+\sqrt{1+0}}2}i\Bigg)=\pm i](https://tex.z-dn.net/?f=%5Csqrt%7B-1%7D%3D%5Cpm%5CBigg%28%5Csqrt%7B%5Cdfrac%7Ba%2B%5Csqrt%7Ba%5E2%2Bb%5E2%7D%7D2%7D%2B%5Csqrt%7B%5Cdfrac%7B-a%2B%5Csqrt%7Ba%5E2%2Bb%5E2%7D%7D2%7Di%5CBigg%29%3D%5C%5C%5C%5C~~~%3D%5Cpm%5CBigg%28%5Csqrt%7B%5Cdfrac%7B-1%2B%5Csqrt%7B1%2B0%7D%7D2%7D%2B%5Csqrt%7B%5Cdfrac%7B1%2B%5Csqrt%7B1%2B0%7D%7D2%7Di%5CBigg%29%3D%5Cpm%20i)
![1)~~\sqrt{-1}=+i;~~~a=0;~b=1\\\\~~\sqrt{i}=\pm\Bigg(\sqrt{\dfrac{0+\sqrt{0+1}}2}+\sqrt{\dfrac{0+\sqrt{0+1}}2}i\Bigg)=\pm \Bigg(\dfrac 1{\sqrt2}+\dfrac 1{\sqrt2}i\Bigg)](https://tex.z-dn.net/?f=1%29~~%5Csqrt%7B-1%7D%3D%2Bi%3B~~~a%3D0%3B~b%3D1%5C%5C%5C%5C~~%5Csqrt%7Bi%7D%3D%5Cpm%5CBigg%28%5Csqrt%7B%5Cdfrac%7B0%2B%5Csqrt%7B0%2B1%7D%7D2%7D%2B%5Csqrt%7B%5Cdfrac%7B0%2B%5Csqrt%7B0%2B1%7D%7D2%7Di%5CBigg%29%3D%5Cpm%20%5CBigg%28%5Cdfrac%201%7B%5Csqrt2%7D%2B%5Cdfrac%201%7B%5Csqrt2%7Di%5CBigg%29)
![2)~~\sqrt{-1}=-i;~~~a=0;~b=-1\\\\~~\sqrt{-i}=\pm\Bigg(\sqrt{\dfrac{0+\sqrt{0+1}}2}-\sqrt{\dfrac{0+\sqrt{0+1}}2}i\Bigg)=\pm \Bigg(\dfrac 1{\sqrt2}-\dfrac 1{\sqrt2}i\Bigg)](https://tex.z-dn.net/?f=2%29~~%5Csqrt%7B-1%7D%3D-i%3B~~~a%3D0%3B~b%3D-1%5C%5C%5C%5C~~%5Csqrt%7B-i%7D%3D%5Cpm%5CBigg%28%5Csqrt%7B%5Cdfrac%7B0%2B%5Csqrt%7B0%2B1%7D%7D2%7D-%5Csqrt%7B%5Cdfrac%7B0%2B%5Csqrt%7B0%2B1%7D%7D2%7Di%5CBigg%29%3D%5Cpm%20%5CBigg%28%5Cdfrac%201%7B%5Csqrt2%7D-%5Cdfrac%201%7B%5Csqrt2%7Di%5CBigg%29)
![\Boldsymbol{\sqrt[4]{-625}=5\sqrt[4]{-1}=5\sqrt{\pm i}=\pm5\Bigg(\dfrac 1{\sqrt2}\pm\dfrac 1{\sqrt2}i\Bigg)=\pm2,5{\sqrt2}\big(1\pm i\big)}](https://tex.z-dn.net/?f=%5CBoldsymbol%7B%5Csqrt%5B4%5D%7B-625%7D%3D5%5Csqrt%5B4%5D%7B-1%7D%3D5%5Csqrt%7B%5Cpm%20i%7D%3D%5Cpm5%5CBigg%28%5Cdfrac%201%7B%5Csqrt2%7D%5Cpm%5Cdfrac%201%7B%5Csqrt2%7Di%5CBigg%29%3D%5Cpm2%2C5%7B%5Csqrt2%7D%5Cbig%281%5Cpm%20i%5Cbig%29%7D)
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В области комплексных чисел (-1) можно представить в тригонометрической форме. Так как r = 1 ,
![-1=\cos\big(\pi+2\pi k\big)+\sin\big(\pi+2\pi k\big)i,~~k\in\mathbb Z](https://tex.z-dn.net/?f=-1%3D%5Ccos%5Cbig%28%5Cpi%2B2%5Cpi%20k%5Cbig%29%2B%5Csin%5Cbig%28%5Cpi%2B2%5Cpi%20k%5Cbig%29i%2C~~k%5Cin%5Cmathbb%20Z)
Тогда для извлечения корня можно использовать формулу Муавра
![\sqrt[4]{-1}=\cos\Bigg(\dfrac{\pi+2\pi k}4\Bigg)+\sin\Bigg(\dfrac{\pi+2\pi k}4\Bigg)i=\\\\~~~~~~~=\cos\Bigg(\dfrac{\pi}4+\dfrac{\pi k}2\Bigg)+\sin\Bigg(\dfrac{\pi}4+\dfrac{\pi k}2\Bigg)i;~~~k=0;1;2;3](https://tex.z-dn.net/?f=%5Csqrt%5B4%5D%7B-1%7D%3D%5Ccos%5CBigg%28%5Cdfrac%7B%5Cpi%2B2%5Cpi%20k%7D4%5CBigg%29%2B%5Csin%5CBigg%28%5Cdfrac%7B%5Cpi%2B2%5Cpi%20k%7D4%5CBigg%29i%3D%5C%5C%5C%5C~~~~~~~%3D%5Ccos%5CBigg%28%5Cdfrac%7B%5Cpi%7D4%2B%5Cdfrac%7B%5Cpi%20k%7D2%5CBigg%29%2B%5Csin%5CBigg%28%5Cdfrac%7B%5Cpi%7D4%2B%5Cdfrac%7B%5Cpi%20k%7D2%5CBigg%29i%3B~~~k%3D0%3B1%3B2%3B3)
![\\\\k=0;~~\cos\bigg(\dfrac{\pi}4\bigg)=\dfrac{\sqrt2}2;~\sin\bigg(\dfrac{\pi}4\bigg)=\dfrac{\sqrt2}2\\\\k=1;~~\cos\bigg(\dfrac{\pi}4+\dfrac{\pi}2\bigg)=-\dfrac{\sqrt2}2;~\sin\bigg(\dfrac{\pi}4+\dfrac{\pi}2\bigg)=\dfrac{\sqrt2}2\\\\k=2;~~\cos\bigg(\dfrac{\pi}4+\pi\bigg)=-\dfrac{\sqrt2}2;~\sin\bigg(\dfrac{\pi}4+\pi\bigg)=-\dfrac{\sqrt2}2\\\\k=3;~~\cos\bigg(\dfrac{\pi}4+\dfrac{3\pi}2\bigg)=\dfrac{\sqrt2}2;~\sin\bigg(\dfrac{\pi}4+\dfrac{3\pi}2\bigg)=-\dfrac{\sqrt2}2](https://tex.z-dn.net/?f=%5C%5C%5C%5Ck%3D0%3B~~%5Ccos%5Cbigg%28%5Cdfrac%7B%5Cpi%7D4%5Cbigg%29%3D%5Cdfrac%7B%5Csqrt2%7D2%3B~%5Csin%5Cbigg%28%5Cdfrac%7B%5Cpi%7D4%5Cbigg%29%3D%5Cdfrac%7B%5Csqrt2%7D2%5C%5C%5C%5Ck%3D1%3B~~%5Ccos%5Cbigg%28%5Cdfrac%7B%5Cpi%7D4%2B%5Cdfrac%7B%5Cpi%7D2%5Cbigg%29%3D-%5Cdfrac%7B%5Csqrt2%7D2%3B~%5Csin%5Cbigg%28%5Cdfrac%7B%5Cpi%7D4%2B%5Cdfrac%7B%5Cpi%7D2%5Cbigg%29%3D%5Cdfrac%7B%5Csqrt2%7D2%5C%5C%5C%5Ck%3D2%3B~~%5Ccos%5Cbigg%28%5Cdfrac%7B%5Cpi%7D4%2B%5Cpi%5Cbigg%29%3D-%5Cdfrac%7B%5Csqrt2%7D2%3B~%5Csin%5Cbigg%28%5Cdfrac%7B%5Cpi%7D4%2B%5Cpi%5Cbigg%29%3D-%5Cdfrac%7B%5Csqrt2%7D2%5C%5C%5C%5Ck%3D3%3B~~%5Ccos%5Cbigg%28%5Cdfrac%7B%5Cpi%7D4%2B%5Cdfrac%7B3%5Cpi%7D2%5Cbigg%29%3D%5Cdfrac%7B%5Csqrt2%7D2%3B~%5Csin%5Cbigg%28%5Cdfrac%7B%5Cpi%7D4%2B%5Cdfrac%7B3%5Cpi%7D2%5Cbigg%29%3D-%5Cdfrac%7B%5Csqrt2%7D2)
![\boldsymbol{\sqrt[4]{-625}=5\sqrt[4]{-1}=\pm5\Bigg(\dfrac {\sqrt2}2\pm\dfrac {\sqrt2}2i\Bigg)=\pm2,5{\sqrt2}(1\pm i)}](https://tex.z-dn.net/?f=%5Cboldsymbol%7B%5Csqrt%5B4%5D%7B-625%7D%3D5%5Csqrt%5B4%5D%7B-1%7D%3D%5Cpm5%5CBigg%28%5Cdfrac%20%7B%5Csqrt2%7D2%5Cpm%5Cdfrac%20%7B%5Csqrt2%7D2i%5CBigg%29%3D%5Cpm2%2C5%7B%5Csqrt2%7D%281%5Cpm%20i%29%7D)