2^333 = (2^3)^111 = 8^111

3^222 = (3^2)^111 = 9 ^111

Vì 8< 9 => 8^111 < 9^111 => 2^333 < 3^222

Ta có: 2^333 = (2^3)^111 = 8^111
3^222 = (3^2)^111 = 9^111
Vì 8^111 < 9^111 nên 2^333 < 3^222

  7^86 + 7^85 - 7^84 

= 7^84 ( 7^2 + 7 - 1 )

=7^84 . ( 49 + 7 - 1 )

= 7^84 ( 56 - 1)

=7^84 . 55 

luôn luôn chia hết cho 55 

Gọi phân số cần tìm là \(\frac{7}{a}\)

Theo bài ra ta có:

\(\frac{-9}{10}

\(\left(\frac{x+4}{2000}+1\right)+\left(\frac{x+3}{2001}+1\right)=\left(\frac{x+2}{2002}+1\right)+\left(\frac{x+1}{2003}+1\right)\)

\(\Rightarrow\frac{x+2004}{2000}+\frac{x+2004}{2001}=\frac{x+2004}{2002}+\frac{x+2004}{2003}\)

\(\Rightarrow\frac{x+2004}{2000}+\frac{x+2004}{2001}-\frac{x+2004}{2002}-\frac{x+2004}{2003}=0\)

\(\Rightarrow\left(x+2004\right)\left(\frac{1}{2000}+\frac{1}{2001}-\frac{1}{2002}-\frac{1}{2003}\right)=0\)

vì \(\frac{1}{2001}+\frac{1}{2002}-\frac{1}{2003}-\frac{1}{2004}\ne0\Rightarrow x+2004=0\Rightarrow x=-2004\)

vậy x=-2004

\(16^7-2^{24}=\left(2^4\right)^7-2^{24}=2^{4.7}-2^{24}=2^{28}-2^{24}=2^{24}.2^4-2^{24}.1=2^{24}\left(2^4-1\right)=2^{24}\left(16-1\right)=15.2^{24}\)

Luôn luôn chia hết cho 15 

    \(16^7-2^{24}\)

\(=268435456-16777216\)

\(=251658240\)

Chia hết cho 15. 

\(251658240:15=16777216\)

Chứng minh định lí hai đường thẳng phân biệt cùng song song với một đường thẳng thứ ba thì chúng song song

\(\frac{x+4}{2011}+\frac{x+3}{2012}=\frac{x+2}{2013}+\frac{x+1}{2014}\)

\(\left(\frac{x+4}{2011}+1\right)+\left(\frac{x+3}{2012}+1\right)=\left(\frac{x+2}{2013}+1\right)+\left(\frac{x+1}{2014}+1\right)\)

\(\frac{x+2015}{2011}+\frac{x+2015}{2012}=\frac{x+2015}{2013}+\frac{x+2015}{2014}\)

\(\frac{x+2015}{2011}+\frac{x+2015}{2012}-\frac{x+2015}{2013}-\frac{x+2015}{2014}=0\)

\(\left(x+2015\right)\left(\frac{1}{2011}+\frac{1}{2012}-\frac{1}{2013}-\frac{1}{2014}\right)=0\)

Vì \(\frac{1}{2011}+\frac{1}{2012}-\frac{1}{2013}-\frac{1}{2014}\ne0\)

nên \(x+2015=0\)

\(x=0-2015\)

\(x=-2015\)

Vậy \(x=-2015\)

\(=-\frac{1}{2}.\left(-\frac{2}{3}\right).\left(-\frac{3}{4}\right)...\left(-\frac{2014}{2015}\right)\)

Số thừa số là (2015 - 2) + 1 = 2014 là số chẵn nên tích sau là số dương và bằng :

\(\frac{1}{2}.\frac{2}{3}.\frac{3}{4}...\frac{2014}{2015}=\frac{1}{2015}\)