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\(\frac{2}{3\cdot4}+\frac{2}{4\cdot5}+...+\frac{2}{82\cdot83}\)
= \(2\left(\frac{1}{3\cdot4}+\frac{1}{4\cdot5}+...+\frac{1}{82\cdot83}\right)\)
\(=2\left(\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{82}-\frac{1}{83}\right)\)
\(=2\left(\frac{1}{3}-\frac{1}{83}\right)\)
\(=2\cdot\frac{80}{249}\)
\(=\frac{160}{249}\)
\(5\dfrac{3}{4}:3+2\dfrac{1}{4}\times\dfrac{1}{3}-\dfrac{1}{8}=\left(5\dfrac{3}{4}+2\dfrac{1}{4}\right)\times\dfrac{1}{3}-\dfrac{1}{8}\\ =8\times\dfrac{1}{3}-\dfrac{1}{8}=\dfrac{8}{3}-\dfrac{1}{8}=\dfrac{61}{24}\\ ----\\ \dfrac{3}{5}:\dfrac{5}{6}:\dfrac{6}{7}:\dfrac{7}{8}+\dfrac{2}{5}+\dfrac{23}{35}\\ =\dfrac{3}{5}\times\dfrac{6}{5}\times\dfrac{7}{6}\times\dfrac{8}{7}+\dfrac{2}{5}+\dfrac{23}{35}=\dfrac{18}{25}\times\dfrac{4}{3}+\dfrac{2}{5}+\dfrac{23}{35}=\dfrac{353}{175}\)
\(4x^2-4x+1=\left(2x\right)^2-2.2x.1+1^2=\left(2x-1\right)^2\\ ---\\ 4x^2-4x-3\\ =4x^2-4x+1-4\\ =\left(2x-1\right)^2-2^2=\left(2x-1-2\right)\left(2x-1+2\right)\\ =\left(2x-3\right)\left(2x+1\right)\)
1: =(2x)^2-2*2x*1+1^2
=(2x-1)^2
2: =4x^2-6x+2x-3
=2x(2x-3)+(2x-3)
=(2x-3)(2x+1)
`@` `\text {Ans}`
`\downarrow`
`2023 \times 14 + 2023 \times 83+97`
`= 2023 \times (14+83) + 97`
`= 2023 \times 97 + 97`
`= 196231 +97`
`= 196328`
`78 \times 31+78 \times 24+78 \times 17+78 \times 22 \times 72`
`= 78 \times (31+24+17+22 \times 72)`
`= 78 \times (31+24+17+1584)`
`= 78 \times 1656`
`= 129168`
`371 \times 69+371 \times 14+629 \times 83`
`= 371 \times (69+14) + 629 \times 83`
`= 371 \times 83 + 629 \times 83`
`= 83 \times (371+629)`
`= 83 \times 1000`
`= 83000`
`91 \times 51+49 \times 163-49 \times 72`
`= 91 \times 51 + 49 \times (163-72)`
`= 91 \times 51+49 \times 91`
`= 91 \times (51+49)`
`= 91 \times 100`
`= 9100`
`3 \times 12 \times 17 + 4 \times 9 \times 81 + 2 \times 6 \times 6`
`= 9 \times 4 \times 17 + 4 \times 9 \times 81 + 4 \times 9 \times 2`
`= 9 \times 4 \times (17+81+2)`
`= 36 \times 100`
`= 3600`
`42 \times 17 + 58 \times 83 \times 42 \times 83 + 42 \times 83 + 17`
`= 17 \times (42+58) + 42 \times 83 \times (58 \times 83 +1)`
`= 17 \times 100 + 3486 \times 4815`
`= 1700 + 16785090`
`= 16786790`
a) 100 - 9 : ( 372 : 3 .y - 1 ) -14 = 83
100 - 9 : ( 372 : 3 . y - 1 ) = 83 + 14
100 - 9 : ( 372 : 3 . y - 1 ) = 97
9 : ( 372 : 3 . y - 1 ) = 100 - 97
9 : ( 372 : 3 . y - 1 ) = 3
372 : 3 . y - 1 = 9 : 3
372 : 3 . y - 1 = 3
372 : 3 . y = 3 + 1
372 : 3 . y = 4
3 . y = 372 : 4
3 . y = 93
y = 93 : 3
y = 31
\(100-9:\left(372:3.y-1\right)-14=83\)
\(100-9:\left(372:3.y-1\right)=97\)
\(9:\left(372:3.y-1\right)=3\)
\(\left(372:3.y-1\right)=3\)
\(372:3.y=4\)
\(3.y=93\)
\(y=31\)
= 3/2
= 6
3/2
6