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a.=[ \(\frac{1}{5}\)x 5 ]5 b. \(\frac{102^3}{40^3}\)=[ \(\frac{102}{40}\)]3 c.=\(\frac{75^{10}.5^{20}}{75^{10}.75^5}\)=\(\frac{5^{20}}{75^5}\)
=15 =[ \(\frac{51}{20}\)]3
=1
\(\frac{45^{10}\times5^{20}}{75^{15}}=\frac{3^{20}\times5^{10}\times5^{20}}{3^{15}\times5^{30}}=3^5=243\)
a. \(\frac{20^5.5^{10}}{100^5}\)
\(=\frac{20^5.\left(5^2\right)^5}{100^5}\)
\(=\frac{20^5.25^5}{100^5}\)
\(=\frac{500^5}{100^5}\)
\(=\left(\frac{500}{100}\right)^5\)
\(=5^5=3125\)
b. \(\frac{\left(0,9\right)^5}{\left(0,3\right)^6}\)
\(=\frac{\left(0,9\right)^5}{\left(0,3\right)^5.0,3}\)
\(=\left(\frac{0,9}{0,3}\right)^5.\frac{1}{0,3}\)
\(=3^5.\frac{1}{0,3}\)
\(=810\)
c. \(\frac{6^3+3.6^2+3^3}{-13}\)
\(=\frac{\left(3.2\right)^3+3.\left(3.2\right)^2+3^3}{-13}\)
\(=\frac{3^3\left(2^3+2^2+1\right)}{-13}\)
\(=\frac{3^3.13}{-13}\)
\(=\left(-3\right)^3\)
\(=-27\)
Bài 1:
a)
\(\frac{8^{20}+4^{20}}{4^{25}+64^5}=\frac{(2^3)^{20}+(2^2)^{20}}{(2^2)^{25}+(2^6)^{5}}=\frac{2^{60}+2^{40}}{2^{50}+2^{30}}=\frac{2^{40}(2^{20}+1)}{2^{30}(2^{20}+1)}=2^{10}\)
b)
\(\frac{45^{10}.5^{20}}{75^{15}}=\frac{(3^2.5)^{10}.5^{20}}{(3.5^2)^{15}}=\frac{3^{20}5^{30}}{3^{15}.5^{30}}=\frac{3^{20}}{3^{15}}=3^5\)
Bài 2:
Ta thấy $(x-2)^{2012}=[(x-2)^{1006}]^2\geq 0$ với mọi $x\in\mathbb{R}$
$|b^2-9|^{2014|\geq 0$ với mọi $b\in\mathbb{R}$ (tính chất trị tuyệt đối)
Do đó để tổng của chúng bằng $0$ thì:
\((x-2)^{2012}=|b^2-9|^{2014}=0\)
\(\Leftrightarrow \left\{\begin{matrix} x-2=0\\ b^2-9=0\end{matrix}\right.\Leftrightarrow \left\{\begin{matrix} x=2\\ b=\pm 3\end{matrix}\right.\)
Vậy.......
a) \(\frac{2^{11}.9^2}{3^5.16^2}=\frac{2^{11}.3^4}{3^5.2^8}=\frac{2^3}{3}=\frac{8}{3}\)
Ta có:
\(A=\frac{3}{1\cdot5}+\frac{3}{5\cdot10}+...+\frac{3}{100\cdot105}\)
\(=\frac{3}{5}\cdot\left(\frac{5}{1\cdot5}+\frac{5}{5\cdot10}+...+\frac{5}{100\cdot105}\right)\)
\(=\frac{3}{5}\cdot\left(1-\frac{1}{5}+\frac{1}{5}-\frac{1}{10}+...+\frac{1}{100}-\frac{1}{105}\right)\)
\(=\frac{3}{5}\left(1-\frac{1}{105}\right)=\frac{3}{5}\cdot\frac{104}{105}=\frac{312}{525}\)
\(A=\frac{25^3.5^5}{6.5^{10}}\)
\(A=\frac{\left(5^2\right)^3.5^5}{6.5^{10}}\)
\(A=\frac{5^6.5^5}{6.5^{10}}\)
\(A=\frac{5^{11}}{6.5^{10}}\)
\(A=\frac{5}{6}\)
(Dùng phương pháp giảm ước)
\(=\frac{\left(5^2\right)^3.5^5}{6.5^{10}}\)
\(=\frac{5^6.5^5}{6.5^{10}}\)
\(=\frac{5^{11}}{6.5^{10}}\)
\(=\frac{5}{6}\)
VẬY \(A=\frac{5}{6}\)