Odpowiedź:
Zad. 1
[tex]a)\\0.125^{-\frac43}=(\frac18)^{-\frac43}=8^{\frac43}=\sqrt[3]{8^4}=\sqrt[3]{8*8*8*8}=8\sqrt[3]8=8\sqrt[3]{2*2*2}=8*2=16\\\\b) \\0.04^{\frac32}=(\frac4{100})^{\frac32}=\sqrt{(\frac{4}{100})^3}=\sqrt{\frac{64}{1000000}}=\frac{8}{1000}=0.008\\\\[/tex]
[tex]c)\\((9+4\sqrt5)^\frac12+(9-4\sqrt5)^\frac12)^2=\\((9+4\sqrt5)^\frac12)^2+2*(9+4\sqrt5)^\frac12(9-4\sqrt5)^\frac12+((9-4\sqrt5)^\frac12)^2=\\9+4\sqrt5+2((9+4\sqrt5)(9-4\sqrt5))^\frac12+9-4\sqrt5=\\18+2(9^2-(4\sqrt5)^2)^\frac12=\\18+2(81-80)^\frac12=\\18+2*1^\frac12=\\18+2*1=\\18+2=20[/tex]
Zad. 2
[tex]a)\\log_{4\sqrt5}(16\sqrt2) \to (4\sqrt5)^x=16\sqrt2 - \text{brak rozwiazan}[/tex]
[tex]b) \\\frac{2log2+2log6}{log4+log3}=\frac{log2^2+log6^2}{log(4*3)}=\frac{log4+log36}{log12}=\frac{log(4*36)}{log12}=\frac{log144}{log12}=\frac{log12^2}{log12}=\frac{2log12}{log12}=2[/tex]
[tex]c)\\36^{1+log_65}=(6^2)^{log_66+log_65}=(6^2)^{log_6(6*5)}=(6^2)^{log_630}=6^{2log_630}=6^{log_6{30^2}}=6^{log_6900}=900[/tex]
Zad. 3
[tex]f(x)=a^x\\A=(\frac12, \sqrt6)\\\sqrt6=a^{\frac12}\\6^{\frac12}=a^\frac12\\a=6\\\\\text{Funkcja rosnaca}[/tex]
Zad. 4
[tex]a)\\f(x)=log_4(2x^2-x-1)\\2x^2-x-1 > 0\\\Delta=(-1)^2-4*2*(-1)=1+8=9\\\sqrt{\Delta}=3\\x_1=\frac{1+3}{4}=\frac{4}4=1\\x_2=\frac{1-3}4=\frac{-2}4=-\frac12\\D \to x\in(-\infty; -\frac12)U(1; \infty)[/tex]
[tex]b)\\f(x)=log_x(4x-3)\\4x-3 > 0\\4x > 3 /:4\\x > \frac34\\x \neq 1 \\D\to x\in(\frac34; 1)U(1; \infty)[/tex]
Zad. 5
[tex]f(x)=log_\frac12(x+1)\\x+1 > 0\\x > -1\\D\to x\in(-1; \infty)\\Zw \to y \in (-\infty; \infty)[/tex]
Szczegółowe wyjaśnienie:
Wzory:
[tex]a^{-\frac{m}n}=\sqrt[n]{\frac{1}{a^m}}\\a^{\frac{m}n}=\sqrt[n]{a^m}\\(a^m)^n=a^{m*n}\\(a+b)(a-b)^2=a^2-b^2\\(a+b)^2=a^2+2ab+b^2\\(a-b)^2=a^2-2ab+b^2\\log_ab=c \to a^c=b\\n*log_ab=log_ab^n\\log_ab+log_ac=log_a(b*c)\\log_ab-log_ac=log_a(\frac{b}c)\\a^{log_ab}=b[/tex]
