9D7Uf
Apskaičiuokite $$\sqrt[6]{14-6\cdot \sqrt {5}}\cdot \sqrt[3]{(3+\sqrt {5})}\cdot \sqrt[3]{2}$$
Sprendimas.
saknis(6,
14- 6* saknis(5))* saknis(3,(3+saknis(5)))* saknis(3,2) =
14- 6* saknis(5))* saknis(3,(3+saknis(5)))* saknis(3,2) = |
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saknis(6,14- 6* saknis(5))* saknis(3,(3+saknis(5)))* saknis(3,2) = $$\sqrt[6]{14-6\cdot \sqrt {5}}\cdot \sqrt[3]{(3+\sqrt {5})}\cdot \sqrt[3]{2}$$ =
14- 6* saknis(5))* saknis(3,(3+saknis(5)))* saknis(3,2) = $$\sqrt[6]{14-6\cdot \sqrt {5}}\cdot \sqrt[3]{(3+\sqrt {5})}\cdot \sqrt[3]{2}$$ = saknis(6,9+5- 6* saknis(5))* saknis(3,(3+saknis(5)))* saknis(3,2) = $$\sqrt[6]{9+5-6\cdot \sqrt {5}}\cdot \sqrt[3]{(3+\sqrt {5})}\cdot \sqrt[3]{2}$$ =
9+5- 6* saknis(5))* saknis(3,(3+saknis(5)))* saknis(3,2) = $$\sqrt[6]{9+5-6\cdot \sqrt {5}}\cdot \sqrt[3]{(3+\sqrt {5})}\cdot \sqrt[3]{2}$$ = saknis(6,9- 6* saknis(5)+5)* saknis(3,(3+saknis(5)))* saknis(3,2) = $$\sqrt[6]{9-6\cdot \sqrt {5}+5}\cdot \sqrt[3]{(3+\sqrt {5})}\cdot \sqrt[3]{2}$$ =
9- 6* saknis(5)+5)* saknis(3,(3+saknis(5)))* saknis(3,2) = $$\sqrt[6]{9-6\cdot \sqrt {5}+5}\cdot \sqrt[3]{(3+\sqrt {5})}\cdot \sqrt[3]{2}$$ = saknis(6, 3^2- 6* saknis(5)+5)* saknis(3,(3+saknis(5)))* saknis(3,2) = $$\sqrt[6]{3^{2}-6\cdot \sqrt {5}+5}\cdot \sqrt[3]{(3+\sqrt {5})}\cdot \sqrt[3]{2}$$ =
3^2- 6* saknis(5)+5)* saknis(3,(3+saknis(5)))* saknis(3,2) = $$\sqrt[6]{3^{2}-6\cdot \sqrt {5}+5}\cdot \sqrt[3]{(3+\sqrt {5})}\cdot \sqrt[3]{2}$$ = saknis(6, 3^2- 2* 3* saknis(5)+5)* saknis(3,(3+saknis(5)))* saknis(3,2) = $$\sqrt[6]{3^{2}-2\cdot 3\cdot \sqrt {5}+5}\cdot \sqrt[3]{(3+\sqrt {5})}\cdot \sqrt[3]{2}$$ =
3^2- 2* 3* saknis(5)+5)* saknis(3,(3+saknis(5)))* saknis(3,2) = $$\sqrt[6]{3^{2}-2\cdot 3\cdot \sqrt {5}+5}\cdot \sqrt[3]{(3+\sqrt {5})}\cdot \sqrt[3]{2}$$ = $${\normalsize 3^{2}-2\cdot 3\cdot \sqrt {5}+5}$$ = $${\normalsize (3-\sqrt {5})^{2}}$$ Paaiškinimas:
Paaiškinimas: Pagal skirtumo kvadrato greitosios daugybos formulę $${\normalsize a^{2}-2\cdot a\cdot b+b^{2} = (a-b)^{2}}$$.
(Čia a =3, b = saknis(5))
(Čia a =3, b = saknis(5))
saknis(6, (3-saknis(5))^2)* saknis(3,(3+saknis(5)))* saknis(3,2) = $$\sqrt[6]{(3-\sqrt {5})^{2}}\cdot \sqrt[3]{(3+\sqrt {5})}\cdot \sqrt[3]{2}$$ =
(3-saknis(5))^2)* saknis(3,(3+saknis(5)))* saknis(3,2) = $$\sqrt[6]{(3-\sqrt {5})^{2}}\cdot \sqrt[3]{(3+\sqrt {5})}\cdot \sqrt[3]{2}$$ = $${\normalsize \sqrt[6]{(3-\sqrt {5})^{2}}}$$ = $${\normalsize \sqrt[3]{3-\sqrt {5}}}$$ Paaiškinimas:
Paaiškinimas: Pagal laispnio šaknies prastinimo formulę $${\normalsize \sqrt[n*k]{a^{(m*k)}} = \sqrt[n]{a^{m}}}$$
k = 2
k = 2
saknis(3,3-saknis(5))* saknis(3,(3+saknis(5)))* saknis(3,2) = $$\sqrt[3]{3-\sqrt {5}}\cdot \sqrt[3]{(3+\sqrt {5})}\cdot \sqrt[3]{2}$$ =
3-saknis(5))* saknis(3,(3+saknis(5)))* saknis(3,2) = $$\sqrt[3]{3-\sqrt {5}}\cdot \sqrt[3]{(3+\sqrt {5})}\cdot \sqrt[3]{2}$$ = $${\normalsize \sqrt[3]{3-\sqrt {5}}\cdot \sqrt[3]{(3+\sqrt {5})}}$$ = $${\normalsize \sqrt[3]{(3-\sqrt {5})\cdot ((3+\sqrt {5}))}}$$ Paaiškinimas:
Paaiškinimas: Pagal to pačio laispnio šaknų sandaugos formulę $${\normalsize \sqrt[n]{a}\cdot \sqrt[n]{b} = \sqrt[n]{a\cdot b}}$$.
saknis(3, (3-saknis(5))* ((3+saknis(5))))* saknis(3,2) = $$\sqrt[3]{(3-\sqrt {5})\cdot ((3+\sqrt {5}))}\cdot \sqrt[3]{2}$$ =
(3-saknis(5))* ((3+saknis(5))))* saknis(3,2) = $$\sqrt[3]{(3-\sqrt {5})\cdot ((3+\sqrt {5}))}\cdot \sqrt[3]{2}$$ = $${\normalsize (3-\sqrt {5})\cdot ((3+\sqrt {5}))}$$ = $${\normalsize (3-\sqrt {5})\cdot (3+\sqrt {5})}$$
saknis(3, (3-saknis(5))* (3+saknis(5)))* saknis(3,2) = $$\sqrt[3]{(3-\sqrt {5})\cdot (3+\sqrt {5})}\cdot \sqrt[3]{2}$$ =
(3-saknis(5))* (3+saknis(5)))* saknis(3,2) = $$\sqrt[3]{(3-\sqrt {5})\cdot (3+\sqrt {5})}\cdot \sqrt[3]{2}$$ = $${\normalsize (3-\sqrt {5})\cdot (3+\sqrt {5})}$$ = $${\normalsize (3^{2}-\sqrt {5}^{2})}$$ Paaiškinimas:
Paaiškinimas: Pagal kvadratų skirtumo formulę
$${\normalsize (a+b)\cdot (a-b) = a^{2}-b^{2}}$$
Čia a = 3, b = saknis(5)
$${\normalsize (a+b)\cdot (a-b) = a^{2}-b^{2}}$$
Čia a = 3, b = saknis(5)
saknis(3,( 3^2- saknis(5)^2))* saknis(3,2) = $$\sqrt[3]{(3^{2}-\sqrt {5}^{2})}\cdot \sqrt[3]{2}$$ =
( 3^2- saknis(5)^2))* saknis(3,2) = $$\sqrt[3]{(3^{2}-\sqrt {5}^{2})}\cdot \sqrt[3]{2}$$ = $${\normalsize \sqrt {5}^{2}}$$ = $${\normalsize 5}$$
saknis(3,( 3^2-5))* saknis(3,2) = $$\sqrt[3]{(3^{2}-5)}\cdot \sqrt[3]{2}$$ =
( 3^2-5))* saknis(3,2) = $$\sqrt[3]{(3^{2}-5)}\cdot \sqrt[3]{2}$$ = $${\normalsize 3^{2}-5}$$ = $${\normalsize 4}$$
saknis(3,(4))* saknis(3,2) = $$\sqrt[3]{(4)}\cdot \sqrt[3]{2}$$ =
(4))* saknis(3,2) = $$\sqrt[3]{(4)}\cdot \sqrt[3]{2}$$ = $${\normalsize (4)}$$ = $${\normalsize 4}$$
saknis(3,4)* saknis(3,2) = $$\sqrt[3]{4}\cdot \sqrt[3]{2}$$ =
4)* saknis(3,2) = $$\sqrt[3]{4}\cdot \sqrt[3]{2}$$ = $${\normalsize \sqrt[3]{4}\cdot \sqrt[3]{2}}$$ = $${\normalsize \sqrt[3]{4\cdot 2}}$$ Paaiškinimas:
Paaiškinimas: Pagal to pačio laispnio šaknų sandaugos formulę $${\normalsize \sqrt[n]{a}\cdot \sqrt[n]{b} = \sqrt[n]{a\cdot b}}$$.
saknis(3, 4* 2) = $$\sqrt[3]{4\cdot 2}$$ =
4* 2) = $$\sqrt[3]{4\cdot 2}$$ = $${\normalsize 4\cdot 2}$$ = $${\normalsize 8}$$
saknis(3,8) = $$\sqrt[3]{8}$$ =
8) = $$\sqrt[3]{8}$$ = $${\normalsize \sqrt[3]{8}}$$ = $${\normalsize 2}$$
2$$2$$
$$\sqrt[6]{14-6\cdot \sqrt {5}}\cdot \sqrt[3]{(3+\sqrt {5})}\cdot \sqrt[3]{2}$$ = $$$$
$$\sqrt[6]{9-6\cdot \sqrt {5}+5}\cdot \sqrt[3]{(3+\sqrt {5})}\cdot \sqrt[3]{2}$$ = $$$$
$$\sqrt[6]{3^{2}-2\cdot 3\cdot \sqrt {5}+5}\cdot \sqrt[3]{(3+\sqrt {5})}\cdot \sqrt[3]{2}$$ = $$$$
$$\sqrt[6]{(3-\sqrt {5})^{2}}\cdot \sqrt[3]{(3+\sqrt {5})}\cdot \sqrt[3]{2}$$ = $$$$
$$\sqrt[3]{3-\sqrt {5}}\cdot \sqrt[3]{(3+\sqrt {5})}\cdot \sqrt[3]{2}$$ = $$$$
$$\sqrt[3]{(3-\sqrt {5})\cdot (3+\sqrt {5})}\cdot \sqrt[3]{2}$$ = $$$$
$$\sqrt[3]{(3^{2}-\sqrt {5}^{2})}\cdot \sqrt[3]{2}$$ = $$$$
$$\sqrt[3]{(3^{2}-5)}\cdot \sqrt[3]{2}$$ = $$$$
$$\sqrt[3]{4}\cdot \sqrt[3]{2}$$ = $$$$
$$\sqrt[3]{8}$$ = $$$$
$$2$$ $$$$
Atsakymas: 2