310 .11 + 310 .5 / 39 . 24
11.911.37-915 / 39.24
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b: \(1024:2^5+140:\left(38+2^5\right)+2\cdot3^2-7^{23}:7^{21}\)
\(=1024:32+140:\left(38+32\right)+2\cdot9-7^2\)
=32+140:70+18-49
=50-49+2
=1+2
=3
c: \(100:\left\lbrace300:\left\lbrack450-\left(4\cdot5^3-2^3\cdot25\right)\right\rbrack\right\rbrace\)
\(=100:\left\lbrace300:\left\lbrack450-\left(4\cdot125-8\cdot25\right)\right\rbrack\right\rbrace\)
\(=100:\left\lbrace300:\left\lbrack450-\left(500-200\right)\right\rbrack\right\rbrace\)
=100:{300:[450-300]}
=100:{300:150}
=100:2
=50
a: 2A=2^2+2^3+...+2^21
=>A=2^21-2
b: B=2+2^2+...+2^100
=>2B=2^2+2^3+...+2^101
=>B=2^101-2
c: C=3+3^2+...+3^10
=>3C=3^2+3^3+...+3^11
=>2C=3^11-3
=>C=(3^11-3)/2
`A = 2 + 2^2 + ... + 2^20`
`=> 2A = 2^2 + 2^3 + ... +2^21`
`=> 2A-A = (2^2 + 2^3 + ... + 2^21) - (2 + 2^2 + ... +2^20)`
`=> A = 2^21 - 2`
`B = 2 + 2^2 + ... + 2^99 + 2^100`
`=>2B = 2^2 + 2^3 + ... + 2^100 + 2^101`
`=> 2B-B = (2^2 + 2^3 + ... + 2^101)- (2 + 2^2 + ... + 2^100)`
`=> B = 2^101 - 2`
`C = 3 + 3^2 + .... + 3^10`
`=>3C = 3^2 + 3^3 + ... +3^11`
`=>3C - C = (3^2 + 3^3 + ... +3^11) - (3 + 3^2 + .... + 3^10)`
`=> 2C = 3^11 - 3`
`=> C = (3^11 - 3)/2
\(27.3^{x-5}=3^{10}\\ \Rightarrow3^3.3^{x-5}=3^{10}\\ \Rightarrow3^{x-2}=3^{10}\\ \Rightarrow x-2=10\\ \Rightarrow x=12\)