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Bài 1 : Bài giải
\(\frac{28^{15}\cdot3^{17}}{84^{16}}=\frac{\left(2^2\cdot7\right)^{15}\cdot3^{17}}{\left(2^2\cdot3\cdot7\right)^{16}}=\frac{2^{30}\cdot7^{15}\cdot3^{17}}{2^{32}\cdot3^{16}\cdot7^{16}}=\frac{3}{2^2\cdot7}=\frac{3}{4\cdot7}=\frac{3}{28}\)
Bài 2 : Bài giải
\(\frac{3^6\cdot21^{12}}{175^9\cdot7^3}=\frac{3^6\cdot\left(3\cdot7\right)^{12}}{\left(5^2\cdot7\right)^9\cdot7^3}=\frac{3^6\cdot3^{12}\cdot7^{12}}{5^{18}\cdot7^9\cdot7^3}=\frac{3^{18}\cdot7^{12}}{5^{18}\cdot7^{12}}=\frac{3^{18}}{5^{18}}\)
\(\frac{3^{10}\cdot6^7\cdot4}{10^9\cdot5^8}=\frac{3^{10}\cdot\left(2\cdot3\right)^7\cdot2^2}{\left(2\cdot5\right)^9\cdot5^8}=\frac{3^{10}\cdot2^7\cdot3^7\cdot2^2}{2^9\cdot5^9\cdot5^8}=\frac{3^{17}\cdot2^9}{2^9\cdot5^{17}}=\frac{3^{17}}{5^{17}}\)
Ta có : \(3^{17}\cdot5^{18}=3^{17}\cdot5^{17}\cdot5=\left(3\cdot5\right)^{17}\cdot5=15^{17}\cdot5\)
\(3^{18}\cdot5^{17}=3\cdot3^{17}\cdot5^{17}=3\cdot\left(3\cdot5\right)^{17}=3\cdot15^{17}\)
\(\text{ Vì }5\cdot15^{17}>3\cdot15^{17}\text{ }\Rightarrow\text{ }3^{17}\cdot5^{18}>3^{18}\cdot5^{17}\text{ }\Rightarrow\text{ }\frac{3^{18}}{5^{18}}< \frac{3^{17}}{5^{17}}\)
Ta có: \(A=\dfrac{3^6.21^{12}}{175^9.7^3}=\dfrac{3^6.3^{12}.7^{12}}{5^{18}.7^9.7^3}=\dfrac{3^{18}.7^{12}}{5^{18}.7^{12}}=\dfrac{3^{18}}{5^{18}}=\left(\dfrac{3}{5}\right)^{18}\)
\(B=\dfrac{3^{10}.6^7.4}{10^9.5^8}=\dfrac{3^{10}.2^7.3^7.2^2}{2^9.5^9.5^8}=\dfrac{3^{17}.2^9}{2^9.5^{17}}=\dfrac{3^{17}}{5^{17}}=\left(\dfrac{3}{5}\right)^{17}\)
Vì \(\left(\dfrac{3}{5}\right)^{18}< \left(\dfrac{3}{5}\right)^{17}\Rightarrow A< B\)
Vậy A < B
Đặt: \(A=\frac{3^6.21^{12}}{175^9.7^3}=\frac{3^{18}.7^{12}}{7^{12}.25^9}=\frac{3^{18}}{5^{18}}=\left(\frac{3}{5}\right)^{18}\)
\(B=\frac{3^{10}.6^7.4}{10^9.5^8}=\frac{3^{10}.2^7.3^7.2^2}{2^9.5^9.5^8}=\frac{3^{17}.2^9}{2^9.5^{17}}=\left(\frac{3}{5}\right)^{17}\)
Vì: \(\left(\frac{3}{5}\right)^{18}< \left(\frac{3}{5}\right)^{17}\Rightarrow A< B\)
4) \(3^{n+2}+3^n=270\)
\(\Rightarrow3^n.3^2+3^n=270\)
\(\Rightarrow3^n.\left(3^2+1\right)=270\)
\(\Rightarrow3^n.\left(9+1\right)=270\)
\(\Rightarrow3^n.10=270\)
\(\Rightarrow3^n=270:10\)
\(\Rightarrow3^n=27\)
\(\Rightarrow3^n=3^3\)
\(\Rightarrow n=3\)
Vậy \(n=3\)
a: \(=\dfrac{3^6\cdot2^{21}}{5^{18}\cdot7^9\cdot7^3}=\dfrac{3^6\cdot2^{21}}{5^{18}\cdot7^{12}}\)
b: \(=\dfrac{3^{10}\cdot3^7\cdot2^7\cdot2^2}{2^9\cdot5^9\cdot5^8}=\dfrac{3^{17}}{5^{17}}\)
`@` `\text {Ans}`
`\downarrow`
\(\dfrac{2^8-2^3}{2^5-1}=\dfrac{2^3\left(2^5-1\right)}{2^5-1}=\dfrac{2^3}{1}=2^3=8\)
_____
\(\dfrac{4^8\cdot9^4}{6^6\cdot8^3}\)
`=`\(\dfrac{\left(2^2\right)^8\cdot\left(3^2\right)^4}{2^6\cdot3^6\cdot\left(2^3\right)^3}\)
`=`\(\dfrac{2^{16}\cdot3^8}{2^6\cdot3^6\cdot2^9}\)
`=`\(\dfrac{2^{16}\cdot3^8}{2^{15}\cdot3^6}\)
`=`\(\dfrac{3^2}{2}\) `=`\(\dfrac{9}{2}\)
______
\(\dfrac{27^4\cdot2^3-3^{10}\cdot4^3}{6^4\cdot9^3}\)
`=`\(\dfrac{\left(3^3\right)^4\cdot2^3-3^{10}\cdot\left(2^2\right)^3}{2^4\cdot3^4\cdot\left(3^2\right)^3}\)
`=`\(\dfrac{3^{12}\cdot2^3-3^{10}\cdot2^6}{2^4\cdot3^4\cdot3^6}\)
`=`\(\dfrac{3^{10}\cdot\left(3^2\cdot2^3-2^6\right)}{3^{10}\cdot2^4}\)
`=`\(\dfrac{72-2^6}{2^4}=\dfrac{8}{16}=\dfrac{1}{2}\)
\(\dfrac{2^8-2^3}{2^5-1}=\dfrac{2^3\left(2^5-1\right)}{2^5-1}=2^3=8\)
\(\dfrac{4^8.9^4}{6^6.8^3}=\dfrac{2^{16}.3^8}{2^6.3^6.2^9}=2.3^2=18\)
\(\dfrac{27^4.2^3-3^{10}.4^3}{6^4.9^3}=\dfrac{3^{12}.2^3-3^{10}.2^6}{2^4.3^4.3^6}=\dfrac{2^3.3^{10}.\left(3^2-2^3\right)}{2^4.3^{10}}=\dfrac{9-8}{2}=\dfrac{1}{2}\)
Đặt\(A=\dfrac{3^6.21^{12}}{175^9.7^3}=\dfrac{3^6.3^{12}.7^{12}}{175^9.7^3}=\dfrac{3^{18}.7^{12}}{\left(5^2\right)^9.7^9.7^3}=\dfrac{3^{18}.7^{12}}{5^{18}.7^{12}}=\dfrac{3^{18}}{5^{18}}=\left(\dfrac{3}{5}\right)^{18}\)
Đặt \(B=\dfrac{3^{10}.6^7.4}{10^9.5^8}=\dfrac{3^{10}.3^7.2^7.2^2}{2^9.5^9.5^8}=\dfrac{3^{17}.2^9}{2^9.5^{17}}=\dfrac{3^{17}}{5^{17}}=\left(\dfrac{3}{5}\right)^{17}\)
Mà \(\left(\dfrac{3}{5}\right)^{18}>\left(\dfrac{3}{5}\right)^{17}\Leftrightarrow A>B\)
\(\Rightarrow\dfrac{3^6.21^{12}}{175^9.7^3}>\dfrac{3^{10}.6^7.4}{10^9.5^8}\)