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Tìm số nguyên a
a)4/5<5/a<10/7 b)2/5<a-1/10<8/15(a-1 là tử, 10 là mẫu)
c)12/7<4/a<8/3. d)5<a^2-15<16
a) \(\dfrac{4}{5}< \dfrac{5}{a}< \dfrac{10}{7}\) \(\left(a\inℤ\right)\)
\(\Leftrightarrow\dfrac{7}{10}< \dfrac{a}{5}< \dfrac{5}{4}\)
\(\Leftrightarrow\dfrac{7.5}{10}< a< \dfrac{5}{4}.5\)
\(\Leftrightarrow\dfrac{7}{2}< a< \dfrac{25}{4}\)
\(\Leftrightarrow a\in\left\{4;5;6\right\}\)
b) \(\dfrac{2}{5}< \dfrac{a-1}{10}< \dfrac{8}{15}\)
\(\Leftrightarrow\dfrac{2.10}{5}< a-1< \dfrac{8.10}{15}\)
\(\Leftrightarrow4< a-1< \dfrac{16}{3}\)
\(\Leftrightarrow5< a< \dfrac{19}{3}\)
\(\Leftrightarrow a\in\left\{6\right\}\)
c) \(\dfrac{12}{7}< \dfrac{4}{a}< \dfrac{8}{3}\)
\(\Leftrightarrow\dfrac{3}{8}< \dfrac{a}{4}< \dfrac{7}{12}\)
\(\Leftrightarrow\dfrac{3.4}{8}< a< \dfrac{7.4}{12}\)
\(\Leftrightarrow\dfrac{3}{2}< a< \dfrac{7}{3}\)
\(\Leftrightarrow a\in\left\{2\right\}\)
d) \(5< a^2-15< 16\)
\(\Leftrightarrow10< a^2< 31\)
\(\Leftrightarrow\sqrt[]{10}< a< \sqrt[]{31}\)
\(\Leftrightarrow a\in\left\{4;5\right\}\)
\(linh_1=\dfrac{8^{10}+4^{10}}{8^4+4^{11}}=\dfrac{\left(2^3\right)^{10}+\left(2^2\right)^{10}}{\left(2^3\right)^4+\left(2^2\right)^{11}}=\dfrac{2^{30}+2^{20}}{2^{12}+2^{22}}=\dfrac{2^{20}\left(2^{10}+1\right)}{2^{12}\left(2^{10}+1\right)}=\dfrac{2^{20}}{2^{12}}=2^8=256\)
\(linh_2=10^3+5.10^2+5^3=\left(5.2\right)^3+5.\left(5.2\right)^2+5^3\)
\(=5^3.2^3+5.5^2.2^2+5^3=5^3.2^3+5^3.2^2+5^3\)
\(=5^3.8+5^3.4+5^3.1=5^3\left(8+4+1\right)=5^3.13=1625=\)
\(a,4^{10}\cdot8^{15}=2^{20}\cdot2^{45}=2^{65}\)
\(b,4^{15}\cdot5^{30}=4^{15}\cdot5^{15}\cdot5^{15}=\left(4\cdot5\cdot5\right)^{15}=100^{15}\)
\(c,27^{16}\cdot9^{10}=3^{48}\cdot3^{20}=3^{68}\)
a) \(4^{10}.8^{15}=\left(2^2\right)^{10}.\left(2^3\right)^{15}=2^{20}.2^{45}=2^{20+45}=2^{65}\)
b) \(4^{15}.5^{30}=\left(2^2\right)^{15}.5^{30}=2^{30}.5^{30}=\left(2.5\right)^{30}=10^{30}\)
c) \(27^{16}.9^{10}=\left(3^3\right)^{16}.\left(3^2\right)^{10}=3^{48}.3^{20}=3^{48+20}=3^{68}\)
\(-\frac{3}{4}< \frac{a}{10}< -\frac{3}{5}\)
\(\Rightarrow\frac{-15}{20}< \frac{2a}{20}< \frac{-12}{20}\)
\(\Rightarrow2a=\left\{-14;-13\right\}\)
\(\Rightarrow a=\left\{-7;-\frac{13}{2}\right\}\)
Chắc là a là số nguyên
\(\Rightarrow a=-7\)