Matematyka
paulinkax04
2017-06-25 17:30:45
Proszę pomocy nie umiem tego zadania
Odpowiedź
Łukasz1994
2017-06-25 19:13:35

zad 6.29 a) 1/(√3 - √2) = (√3 + √)/[(√3 - √2)(√3 + √2)] = (√3 + √2)/(3 - 2) = (√3 + √2)/1 =  = √3 + √2 b) 1/(√6 + √5) = (√6 - √5)/[(√6 + √5)(√6 - √5)] = (√6 - √5)/(6 - 5) = (√6 - √5)/1 =  = √6 - √5 c) 1/(√a + √b)   dla a > 0 i b > 0 i a - b = 1 1/(√a + √b) = (√a - √b)/[(√a + √b)(√a - √b)] = (√a - √b)/(a - b) = (√a - √b)/1 =  = √a - √b d) 1/(√a - √b)   dla a > 0 , b > 0 i a - b = 1 1/(√a + √b) = (√a + √b)/[(√a - √b)(√a + √b)] = (√a + √b)/(a - b) = (√a + √b)/1 =  = √a + √b zad 6.30 m = 3 + √5 , n = 3 - √5 a) m² = (3 + √5)² =9 + 6√5 + 5 = 14 + 6√5  b) n² = (3 - √5)² = 9 - 6√5 + 5 = 14 - 6√5  c) m² -n² = 14 + 6√5 - (14 - 6√5) = 14 + 6√5 - 14 + 6√5 = 12√5 d) (m - n)² = [(3 + √5) - (3 - √5)]² = (3 + √5 - 3 + √5)² = (2√5)² = 4 *5 = 20 e) (m + n)² = (3 + √5 + 3 - √5)² = 6² = 36 zad 6.31 a) (7 + 2√6)² = 49 + 28√6 + 24 = 73 + 28√6 b) (4 - 2√3)² = 16 - 16√3 + 12 = 28 - 16√3 c) (9 - 4√5)² = 81 - 72√5 + 80 = 161 + 72√5 d) (5 + 2√6)² = 25 + 20√6 + 24 = 49 + 20√6 zad 6.32 a) [√(9 + 4√5)]² = (√5 + 2)² 9 + 4√5 = 5 + 4√5 + 4 9 + 4√5 = 9 + 4√5 L = P b) [√(8 + 2√7)]² = (√7 + 1)² 8 + 2√7 = 7 + 2√7 + 1 8 + 2√7 = 8 + 2√7 L = P

dziuska94
2017-06-25 19:14:50

[latex]6.29\\a) frac{1}{sqrt{3}-sqrt{2}}cdotfrac{sqrt{3}+sqrt{2}}{sqrt{3}+sqrt{2}} = frac{sqrt{3}+sqrt{2}}{(sqrt{3})^{2}-(sqrt{2})^{2}} = frac{sqrt{3}+sqrt{2}}{3-2} = frac{sqrt{3}+sqrt{2}}{1} = sqrt{3}+sqrt{2}\\b) frac{1}{sqrt{6}+sqrt{5}}cdotfrac{sqrt{6}-sqrt{5}}{sqrt{6}-sqrt{5}} = frac{sqrt{6}-sqrt{5}}{6-5} = frac{sqrt{6}-sqrt{5}}{1} = sqrt{6}-sqrt{5}[/latex] [latex]c) frac{1}{sqrt{a}+sqrt{b}}cdotfrac{sqrt{a}-sqrt{b}}{sqrt{a}-sqrt{b}}=frac{sqrt{a}-sqrt{b}}{a-b} = frac{sqrt{a}-sqrt{b}}{1} = sqrt{a}-sqrt{b}\\d) frac{1}{sqrt{a}-sqrt{b}}cdotfrac{sqrt{a}+sqrt{b}}{sqrt{a}+sqrt{b}} = frac{sqrt{a}+sqrt{b}}{a-b}=frac{sqrt{a}+sqrt{b}}{1} = sqrt{a}+sqrt{b}[/latex] [latex]6.30\a) m^{2} = (3+sqrt{5})^{2} = 3^{2}+2cdot3cdotsqrt{5} + (sqrt{5})^{2} = 9+6sqrt{5}+5 = 14+6sqrt5[/latex] [latex]b) n^{2} = (3-sqrt{5})^{2} = 9-6sqrt{5}+5 = 14-6sqrt{5}\\c) m^{2}-n^{2} = 14+6sqrt{5}-(14-6sqrt{5}) = 14+6sqrt{5}-14+6sqrt{5} = 12sqrt{5}\\d) (m-n)^{2} = ((3+sqrt{5}-(3-sqrt{5}))^{2} = (3+sqrt{5}-3+sqrt{5})^{2} =\=(2sqrt{5})^{2}=2^{2}cdot(sqrt{5})^{2} = 4cdot5 = 20\\e) (m+n)^{2} = (3+sqrt{5}+3-sqrt{5})^{2} = 6^{2} = 36 [/latex] 6.31 (a + b)² = a² + 2ab + b² (a - b)² = a² - 2ab + b² a) 2ab = 2√6      /:2 ab = √6    ⇒  a = 1,  b = √6 7 + 2√6 = (1 + √6)² b) 2ab = 2√3     /:2 ab = √3     ⇒  a = 1,  b = √3 4 - 2√3 = (1 - √3)² c) 2ab = 4√5     /:2 ab = 2√5     ⇒  a = 2,  b = √5 9 - 4√5 = (2 - √5)² d) 2ab = 2√6     /:2 ab = √6 = √3 · √2 a = √3,  b = √2 5 + 2√6 = (√3 + √2)² [latex]6.32\a)\sqrt{9+4sqrt{5}} = sqrt{5+4sqrt{5}+4} = sqrt{(sqrt{5}+2)^{2}} = |sqrt{5}+2| = sqrt{5}+2\\b)\sqrt{8+2sqrt{7}}=sqrt{7+2sqrt{7}+1}} = sqrt{(sqrt7+1)^{2}} =|sqrt{7}+1| = sqrt{7}+1[/latex]

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