Rút gọn biểu thức sau
B = 1 + \(\frac{3}{2^3}+\frac{4}{2^4}+\frac{5}{2^5}+...+\frac{200}{2^{200}}\)
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Rút gọn biểu thức sau
B = 1 + \(\frac{3}{2^3}+\frac{4}{2^4}+\frac{5}{2^5}+...+\frac{200}{2^{200}}\)
9 tháng 8 2018 lúc 11:25
Rút gọn biểu thức:
a) \(\left(\frac{1}{2}\sqrt{\frac{1}{2}}-\frac{3}{2}.\sqrt{2}+\frac{4}{5}.\sqrt{200}\right):\frac{1}{8}\)
\(\left(\frac{1}{2}\sqrt{\frac{1}{2}}-\frac{3}{2}\sqrt{2}+\frac{4}{5}\sqrt{200}\right):\frac{1}{8}\)
\(=8.\left(\frac{1}{\sqrt{8}}-\frac{3}{\sqrt{2}}+8\sqrt{2}\right)\)
\(=2\sqrt{2}-12\sqrt{2}+64\sqrt{2}=54\sqrt{2}\)
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Rút gọn các biểu thức sau :
\(\frac{1}{3}-\frac{1}{3^2}+\frac{1}{3^3}-...-\frac{1}{3^{198}}+\frac{1}{3^{199}}\)
\(\frac{2}{5^1}+\frac{3}{5^2}+\frac{4}{5^3}+...+\frac{100}{5^{99}}\)
\(\frac{1}{2^2-1}+\frac{1}{3^2-1}+\frac{1}{4^2-1}+...+\frac{1}{200^2-1}\)
a, Dat A =\(\frac{1}{3}-\frac{1}{3^2}+\frac{1}{3^3}-...-\frac{1}{3^{198}}+\frac{1}{3^{199}}\)
\(\Rightarrow\frac{1}{3}A=\frac{1}{3^2}-\frac{1}{3^3}+\frac{1}{3^4}-...-\frac{1}{3^{199}}+\frac{1}{3^{200}}\)
\(\Rightarrow\frac{1}{3}A+A=\left(\frac{1}{3^2}-\frac{1}{3^3}+\frac{1}{3^4}-...-\frac{1}{3^{199}}+\frac{1}{3^{200}}\right)+\left(\frac{1}{3}-\frac{1}{3^2}+\frac{1}{3^3}-...-\frac{1}{3^{198}}+\frac{1}{3^{199}}\right)\)
\(\Rightarrow\frac{4}{3}A=\frac{1}{3}+\frac{1}{3^{200}}\)
\(\Rightarrow A=\frac{\frac{1}{3}+\frac{1}{3^{200}}}{\frac{4}{3}}\)
chung minh tuong tu cau b va c
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Rút gọn biểu thức sau :
\(A=\frac{4+\sqrt{3}}{\sqrt{1}+\sqrt{3}}+\frac{6+\sqrt{8}}{\sqrt{3}+\sqrt{5}}+...+\frac{2k+\sqrt{k^2-1}}{\sqrt{k-1}+\sqrt{k+1}}+....+\frac{200+\sqrt{999}}{\sqrt{99}+\sqrt{100}}\)
Với mọi \(k\ge2\) thì \(\frac{2k+\sqrt{k^2-1}}{\sqrt{k-1}+\sqrt{k+1}}=\frac{\left[\left(\sqrt{k-1}\right)^2+\left(\sqrt{k+1}\right)^2+\sqrt{\left(k-1\right)\left(k+1\right)}\right]\left(\sqrt{k+1}-\sqrt{k-1}\right)}{\left(\sqrt{k-1}+\sqrt{k+1}\right)\left(\sqrt{k+1}-\sqrt{k-1}\right)}\)
\(=\frac{\sqrt{\left(k+1\right)^3}-\sqrt{\left(k-1\right)^3}}{2}\)
Suy ra tổng đã cho có thể viết là :
\(A=\frac{1}{2}\left[\sqrt{3^3}-\sqrt{1^3}+\sqrt{4^3}-\sqrt{2^3}+\sqrt{5^3}-\sqrt{3^3}+\sqrt{6^3}-\sqrt{4^3}+...+\sqrt{101^3}-\sqrt{99^3}\right]\)
\(=\frac{1}{2}\left[-1-\sqrt{2^3}+\sqrt{101^3}+\sqrt{100^3}\right]\)
\(=\frac{999+\sqrt{101^3}-\sqrt{8}}{2}\)
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Rút gọn biểu thức:(\(\frac{1}{2}\sqrt{\frac{1}{2}}-\frac{3}{2}\sqrt{2}+\frac{4}{5}\sqrt{200}\)):\(\frac{1}{8}\)
=(\(\frac{1}{2}\sqrt{\frac{2}{2^2}}-\frac{3}{2}\sqrt{2}+\frac{4}{5}\sqrt{10^2.2}\)) .8
=(\(\frac{1}{2}\frac{\sqrt{2}}{2}-\frac{3\sqrt{2}}{2}+\frac{4.10\sqrt{2}}{5}\)).8
=(\(\frac{\sqrt{2}}{2}-\frac{3\sqrt{2}}{2}+8\sqrt{2}\)).8
=\(2\sqrt{2}-12\sqrt{2}+64\sqrt{2}\)
=\(54\sqrt{2}\)
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Rút gọn các biểu thức sau:
1) A=\(\frac{4^59^4-2.6^9}{2^{10}3^8+6^8.20}\)
2) B=\((\frac{3}{5})^2.5^2-(2\frac{1}{4})^3:\left(\frac{3}{4}\right)^3+\frac{1}{2}\)
1) A = -1 / 3
2) B = -35 / 2
Chúc em làm bài tốt !
Rút gọn các biểu thức sau:\(\frac{\frac{1}{2}-\frac{1}{3}+\frac{2}{5}+\frac{1}{8}}{\frac{1}{6}+\frac{3}{20}-\frac{1}{3}+\frac{3}{4}}\)
Rút gọn các biểu thức sau :
A=\(\frac{2.8^4.27^2+4.6^9}{2^7.6^7+2^7.40.9^4}\)
B=\(\frac{\left(\frac{2}{3}\right)^3.\left(-\frac{3}{4}\right)^2.\left(-1\right)^5}{\left(\frac{2}{5}\right)^2.\left(-\frac{5}{12}\right)^3}\)
Rút gọn biểu thức :
B=\(\left(\frac{3}{5}\right)^2.5^2-\left(2\frac{1}{4}\right)^3:\left(\frac{3}{4}\right)^3+\frac{1}{2}\)
\(B=\left(\frac{3}{5}\right)^2\cdot5^2-\left(2\frac{1}{4}\right)^3:\left(\frac{3}{4}\right)^3+\frac{1}{2}\)
\(B=\left(\frac{3}{5}\cdot5\right)^2-\left(\frac{9}{4}:\frac{3}{4}\right)^3+\frac{1}{2}\)
\(B=3^2-\left(\frac{9}{4}\cdot\frac{4}{3}\right)^3+\frac{1}{2}\)
\(B=3^2-3^3+\frac{1}{2}=-18+\frac{1}{2}=-\frac{35}{2}\)