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AH
Akai Haruma
Giáo viên
9 tháng 7 2018

a) \(2^{2014}\)\(3^{1343}\)

Ta có:

\(2^{2014}=(2^3)^{\frac{2014}{3}}=8^{\frac{2014}{3}}< 9^{\frac{2014}{3}}\)

\(3^{1343}=(3^2)^{\frac{1343}{2}}=9^{\frac{1343}{2}}> 9^{\frac{2014}{3}}\)

\(\rightarrow 2^{2014}< 3^{1343}\)

b) \(31^{11}\)\(17^{44}\)

Có: \(17^{44}=(17^4)^{11}> (17.2)^{11}>31^{11}\)

AH
Akai Haruma
Giáo viên
9 tháng 7 2018

c)

\(A=\frac{1}{2^1}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{50}}\)

\(\Rightarrow 2A=1+\frac{1}{2^1}+\frac{1}{2^2}+..+\frac{1}{2^{49}}\)

Lấy vế sau trừ vế trước thu được:

\(2A-A=1-\frac{1}{2^{50}}< 1\)

\(\Leftrightarrow A< 1\)

d) \(B=\frac{1}{3^1}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^{100}}\)

\(\Rightarrow 3B=1+\frac{1}{3^1}+\frac{1}{3^2}+...+\frac{1}{3^{99}}\)

Lấy vế sau trừ vế trước:

\(\Rightarrow 3B-B=1-\frac{1}{3^{100}}< 1\)

\(\Leftrightarrow 2B< 1\Rightarrow B< \frac{1}{2}\)

AH
Akai Haruma
Giáo viên
9 tháng 7 2018

b) \(B=\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)....\left(1-\frac{1}{2018}\right)\)

\(=\frac{2-1}{2}.\frac{3-1}{3}.\frac{4-1}{4}....\frac{2018-1}{2018}\)

\(=\frac{1}{2}.\frac{2}{3}.\frac{3}{4}...\frac{2017}{2018}=\frac{1.2.3...2017}{2.3.4...2018}=\frac{1}{2018}\)

c) Giữa các biểu thức là dấu nhân hay dấu cộng vậy bạn?

AH
Akai Haruma
Giáo viên
9 tháng 7 2018

d)

\(D=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{99.100}\)

\(D=\frac{2-1}{1.2}+\frac{3-2}{2.3}+\frac{4-3}{3.4}+...+\frac{100-99}{99.100}\)

\(=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}\)

\(=1-\frac{1}{100}=\frac{99}{100}\)

e) \(E=\frac{1}{3.5}+\frac{1}{5.7}+\frac{1}{7.9}+...+\frac{1}{97.99}\)

\(2E=\frac{2}{3.5}+\frac{2}{5.7}+\frac{2}{7.9}+...+\frac{2}{97.99}\)

\(2E=\frac{5-3}{3.5}+\frac{7-5}{5.7}+\frac{9-7}{7.9}+....+\frac{99-97}{97.99}\)

\(2E=\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+...+\frac{1}{97}-\frac{1}{99}\)

\(=\frac{1}{3}-\frac{1}{99}=\frac{32}{99}\)

\(\Rightarrow E=\frac{16}{99}\)

Bài 1: 

a: \(=\dfrac{-1}{8}+1-\dfrac{9}{4}-1\)

\(=\dfrac{-1}{8}-\dfrac{18}{8}=\dfrac{-19}{8}\)

b: \(=4\cdot1-2\cdot\dfrac{1}{4}+3\cdot\dfrac{-1}{2}+1\)

\(=4-\dfrac{1}{2}-\dfrac{3}{2}+1\)

=5-2

=3

5 tháng 7 2017

1) Tính

a) 253 : 52 = (52)3 : 52 = 56 : 52 = 54 = 625

\(b)\left(\dfrac{3}{7}\right)^{21}:\left(\dfrac{9}{49}\right)^6=\left(\dfrac{3}{7}\right)^{21}:\left[\left(\dfrac{3}{7}\right)^2\right]^6=\left(\dfrac{3}{7}\right)^{21}:\left(\dfrac{3}{7}\right)^{12}=\left(\dfrac{3}{7}\right)^9\) d) 9 . 32 . \(\dfrac{1}{81}\) . 32 = 32 . 32 . \(\dfrac{1}{3^4}\) . 32 = 9

2) Tìm x thuộc Q, biết:

a) 3x + 2 = 27

=> 3x + 2 = 33

x + 2 = 3

x = 3 - 2

x = 1

b) \(\left(\dfrac{1}{2}x-3\right)^4=81\)

\(\Rightarrow\left(\dfrac{1}{2}x-3\right)^4=3^4\)

\(\dfrac{1}{2}x-3=3^{ }\)

\(\dfrac{1}{2}x=3+3\)

\(\dfrac{1}{2}x=9\)

\(x=9:\dfrac{1}{2}\)

\(x=18\)

c) \(\left(x-\dfrac{1}{2}\right)^3=-27\)

\(\Rightarrow\left(x-\dfrac{1}{2}\right)^3=\left(-3\right)^3\)

\(x-\dfrac{1}{2}=-3\)

\(x=-3+\dfrac{1}{2}\)

\(x=\dfrac{-5}{2}\)

d) 5 . 5x + 1 = 125

5x + 1 = 125 : 5

5x + 1 = 25

5x + 1 = 52

x + 1 = 2

x = 2 - 1

x = 1.

16 tháng 10 2018

b) \(\dfrac{7}{15}-\dfrac{9}{19}\)\(-\dfrac{-8}{15}-\dfrac{10}{19}\)
=\(\left(\dfrac{7}{15}-\dfrac{8}{15}\right)\) \(-\left(\dfrac{9}{19}-\dfrac{10}{19}\right)\)
= \(-\dfrac{1}{15}\) - \(\left(-\dfrac{1}{19}\right)\)
\(=-\dfrac{1}{15}\) + \(\dfrac{1}{19}\)

= \(-\dfrac{4}{285}\)

16 tháng 10 2018

c) \(1\dfrac{1}{3}\) \(\div\) \(\dfrac{4}{5}\) + 2\(\dfrac{2}{3}\) \(\div\)\(\dfrac{4}{5}\)
= \(\left(1\dfrac{1}{3}+2\dfrac{2}{3}\right)\) \(\div\dfrac{4}{5}\)
= \(\left[\left(1+2\right)+\left(\dfrac{1}{3}+\dfrac{2}{3}\right)\right]\) \(\div\dfrac{4}{5}\)
= ( 3 + 1 ) \(\div\dfrac{4}{5}\)
= 4 \(\div\dfrac{4}{5}\)
= \(\dfrac{4.5}{4}\)
= 5

a: \(\dfrac{3x+2}{5x+7}=\dfrac{3x-1}{5x+1}\)

\(\Leftrightarrow\left(3x+2\right)\left(5x+1\right)=\left(3x-1\right)\left(5x+7\right)\)

\(\Leftrightarrow15x^2+3x+10x+2=15x^2+21x-5x-7\)

=>16x-7=13x+2

=>3x=9

hay x=3

b: \(\dfrac{x+1}{2016}+\dfrac{x}{2017}=\dfrac{x+2}{2015}+\dfrac{x+3}{2014}\)

\(\Leftrightarrow\left(\dfrac{x+1}{2016}+1\right)+\left(\dfrac{x}{2017}+1\right)=\left(\dfrac{x+2}{2015}+1\right)+\left(\dfrac{x+3}{2014}+1\right)\)

=>x+2017=0

hay x=-2017

e: \(\left(2x-3\right)^2=144\)

=>2x-3=12 hoặc 2x-3=-12

=>2x=15 hoặc 2x=-9

=>x=15/2 hoặc x=-9/2

14 tháng 6 2017

a.

| x | = 5,6

=>\(\left[{}\begin{matrix}x=5,6\\x=-5,6\end{matrix}\right.\)

Vậy \(x\in\left\{-5,6;5,6\right\}\)

b, \(\left|x-3,5\right|=5\)

=>\(\left[{}\begin{matrix}x-3,5=5\\x-3,5=-5\end{matrix}\right.\)

=>\(\left[{}\begin{matrix}x=8,5\\x=-1,5\end{matrix}\right.\)

Vậy \(x\in\left\{-1,5;8,5\right\}\)

c,\(\left|x-\dfrac{3}{4}\right|-\dfrac{1}{2}=0\)

=> \(\left|x-\dfrac{3}{4}\right|=\dfrac{1}{2}\)

=>\(\left[{}\begin{matrix}x-\dfrac{3}{4}=\dfrac{1}{2}\\x-\dfrac{3}{4}=-\dfrac{1}{2}\end{matrix}\right.\)

=> \(\left[{}\begin{matrix}x=\dfrac{5}{4}\\x=\dfrac{1}{4}\end{matrix}\right.\)

Vậy \(x\in\left\{\dfrac{1}{4};\dfrac{5}{4}\right\}\)

d,\(\left|4x\right|-\left(\left|-13,5\right|\right)=\left|\dfrac{1}{4}\right|\)

=> \(\left|4x\right|-13,5=\dfrac{1}{4}\)

=> \(\left|4x\right|=13,75\)

=>\(\left[{}\begin{matrix}4x=13,75\\4x=-13,75\end{matrix}\right.\)

=>\(\left[{}\begin{matrix}x=3,4375\\x=-3,4375\end{matrix}\right.\)

Vậy \(x\in\left\{-3,4375;3,4375\right\}\)

14 tháng 6 2017

e, ( x - 1 ) 3 = 27

=> x - 1 = 3

=> x = 4

Vậy x = 4

f, ( 2x - 3)2 = 36

=> \(\left[{}\begin{matrix}2x-3=6\\2x-3=-6\end{matrix}\right.\)

=> \(\left[{}\begin{matrix}x=4,5\\x=-1,5\end{matrix}\right.\)

Vậy x\(\in\left\{-1,5;4,5\right\}\)

g, \(5^{x+2}=625\)

=> \(5^{x+2}=5^4\)

=> x + 2 = 4

=> x = 2

Vậy x = 2

h, ( 2x - 1)3 = -8

=> 2x - 1 = -2

=> x = \(\dfrac{-1}{2}\)

Vậy x = \(\dfrac{-1}{2}\)

i, \(\dfrac{1}{4}.\dfrac{2}{6}.\dfrac{3}{8}.\dfrac{4}{10}.\dfrac{5}{12}...\dfrac{30}{62}.\dfrac{31}{64}=2^x\)

=> \(\dfrac{1.2.3.4.5...30.31}{4.6.8.10.12...62.64}=2^x\)

=>\(\dfrac{1.2.3.4.5...30.31}{\left(2.3.4.5...30.31.32\right)\left(2.2.2.2...2.2_{ }\right)}=2^x\)(có 31 số 2)

=> \(\dfrac{1}{32.2^{31}}=2^x\)

=> \(\dfrac{1}{2^{36}}=2^x\)

=> x = -36

Vậy x = -36

3 tháng 8 2018

Bài 1:

\(a)\dfrac{20^5.5^{10}}{100^5}=\dfrac{20^5.5^5.5^5}{100^5}=\dfrac{100^5.3125}{100^5}=3125\)

3 tháng 8 2018

2.

a)A có 36 sô hạng , chia A thành 18 nhóm , mỗi nhóm có 2 số hạng .

Ta có : A = \(\left(3+3^2\right)+\left(3^3+3^4\right)+....+\left(3^{35}+3^{36}\right)\)

\(A=3.\left(1+3\right)+3^3.\left(1+3\right)+...+3^{35}.\left(1+3\right)\)

\(A=3.4+3^3.4+...+3^{35}.4\)

\(A=4.\left(3+3^3+...+3^{35}\right)\)

Vậy A chia hết cho 4 .

b)Chia A thành 13 nhóm mỗi nhóm có 3 số hạng

Ta có : \(A=\left(3+3^2+3^3\right)+...+\left(3^{34}+3^{35}+3^{36}\right)\)

\(A=3.\left(1+3+9\right)+...+3^{34}.\left(1+3+9\right)\)

A=\(3.13+...+3^{34}.13\)

A= \(13.\left(3+..+3^{34}\right)\)

Vậy A chia hết cho 13

c) Tương tự như câu a và câu b

16 tháng 10 2022

a: \(\Leftrightarrow4^x\left(\dfrac{3}{2}+\dfrac{5}{3}\cdot4^2\right)=4^8\left(\dfrac{3}{2}+\dfrac{5}{3}\cdot4^2\right)\)

=>4^x=4^8

=>x=8

b: \(\Leftrightarrow2^x\cdot\dfrac{1}{2}+2^x\cdot2=2^{10}\left(2^2+1\right)\)

=>2^x=2^11

=>x=11

c: =>1/6*6^x+6^x*36=6^15(1+6^3)

=>6^x=6*6^15

=>x=16

d: \(\Leftrightarrow8^x\left(\dfrac{5}{3}\cdot8^2-\dfrac{3}{5}\right)=8^9\left(\dfrac{5}{3}\cdot8^2-\dfrac{3}{5}\right)\)

=>x=9

23 tháng 9 2017

a.\(3^{x-1}=243\)

\(3^x:3^1=243\)

\(3^x=729\)

\(\Leftrightarrow3^6=729\)

\(\Leftrightarrow x=6\)

b.\(\left(\dfrac{2}{3}\right)^{x+1}=\dfrac{8}{4}\)

\(\left(\dfrac{2}{3}\right)^x.\left(\dfrac{2}{3}\right)=\dfrac{8}{4}\)

\(\left(\dfrac{2}{3}\right)^x=3\)

Câu b tính đến đây rồi không mò đc x nữa.