So sánh A=2^0+2^1+2+2+2^3 + 2^2010 và B=2 2011 1

Đáp án + giải thích các bước giải:

`a, A = 2^0 + 2^1 + 2^2 + .... + 2^2020`

  `2A = 2 . (2^0 + 2^1 + 2^2 + ... + 2^2020)`

  `2A = 2^1 + 2^2 + 2^3 + .... + 2^2021`

` 2A - A = (2^1 + 2^2 + 2^3 + .... + 2^2021) - (2^0 + 2^1 + 2^2 + .... + 2^2020)`

` A = 2^1 + 2^2 + 2^3 + .... + 2^2021 - 1 - 2^1 - 2^2 - .... - 2^2020`

` A = 2^2021 - 1`

Mà `B = 2^2021 - 1` nên `A = B`

`b,` Ta có:

`A = 2009 . 2011`

   ` = (2010 - 1) .  2011`

`    = 2010 . 2011 - 1 . 2011`

`    = 2010 . 2011 - 2011`

`B = 2010^2`

   `= 2010 . 2010`

   ` = (2011 - 1) . 2010`

   ` = 2011 . 2010 - 1 . 2010`

    ` = 2011 . 2010 - 2010`

Vì `2011 . 2010 > 0`

`=> 2011 . 2010 - 2010 > 2011 . 2010 - 2011`

`=> 2009 . 2011 < 2010^2`

`=> A < B`

`c,` Ta có:

`10^30 = (10^3)^10 = 1000^10`

`2^100 = (2^10)^10 = 1024^10`

Mà  `1024 > 1000` nên `1024^10 > 1000^10`

Vậy `2^100 > 10^30`

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Tính chất :

`a . (b - 1) = a . b - a`

`(a^b)^c = a^(b . c)`

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\(A=2^0+2^1+2^2+2^3+-+2^{2010}\)

\(2A=2^1+2^2+2^3+2^4+-.+2^{2011}\)

\(2A-A=\left(2^1+2^2+2^3+2^4+-.+2^{2011}\right)-\left(2^0+2^1+2^2+2^3+-+2^{2010}\right)\)\(A=2^{2011}-2^0=2^{2011}-1\)

\(A=B=2^{2011}-1\)

\(A=2009.2011=2009.\left(2010+1\right)=2009.2010+2011\)

\(B=2010^2=2010.2010=2010\left(2009+1\right)=2010.2009+2010\)

\(A>B\)

\(A=10^{30}=\left(10^3\right)^{10}=1000^{10}\)

\(B=2^{100}=\left(2^{10}\right)^{10}=1024^{10}\)

\(A< B\)

\(A=333^{444}=\left(333^4\right)^{111}=\left(3^4.111^4\right)^{111}=\left(81.111^4\right)^{111}\)

\(B=444^{333}=\left(444^3\right)^{111}=\left(4^3.111^3\right)^{111}=\left(64.111^3\right)^{111}\)

\(A>B\)

\(A=3^{150}\)

\(B=5^{300}=\left(5^2\right)^{150}=25^{150}\)

\(A< B\)

Hay nhất

A = 2^0 +2^1 + 2^2 + 2^3 +...+2^2010 và B =2^2011 - 1

Ta có :

2A =2^1 +2^2 + 2^3 + 2^4 +...+2^2011

2A - A =( 2^1 +2^2 + 2^3 + 2^4 +...+2^2011 ) -( 1 + 2^2 + 2^3 +...+2^2010)

A =2^2011 - 1

MàB =2^2011 - 1

⇒ A = B

Bài viết tham gia Hoa điểm 10 2016

So sánh

a) 2^0 + 2^1 + 2^2 + 2^3 +...+2^2010 Và B = 2^2011 - 1

b) A = 2009 . 2011 và B = 2010^2