Simplifier les écritures suivantes :

` A = ln((4sqrt(2))/(27)) +ln(12sqrt(3)) +ln(3sqrt(3))`

` B = 7/(16)ln(3+ 2sqrt(2)) -4ln(1+sqrt(2)) -(25)/8ln(sqrt(2) -1) `

`D = 2ln(esqrt(e)) -3ln(e^3) -2ln(e^(-6)) `

`E = ln(1/2) +ln(2/3) + ..... ln((98)/(99)) + ln((99)/(100)) `

`F = 3/8ln(7+4sqrt(3)) -ln(2+sqrt(3)) -1/4 ln(2-sqrt(3)) `

Simplifier ` A = ln((4sqrt(2))/(27)) +ln(12sqrt(3)) +ln(3sqrt(3))`

On a ` A = ln((4sqrt(2))/(27)) +ln(12sqrt(3)) +ln(3sqrt(3))`

` = ln ( (4sqrt(2)xx12sqrt(3)xxsqrt(3) xx3 )/(27)) `

` = ln ( ( 4xx4xx3 xx9sqrt(2))/(27)) `

` = ln(16) +ln(sqrt(2)) = ln(2^4) +1/2ln2 = 4ln(2) +1/2ln2 `

` = (9ln2)/2 `

`=> A= (9ln2)/2 `

` B = 7/(16)ln(3+ 2sqrt(2)) -4ln(1+sqrt(2)) -(25)/8ln(sqrt(2) -1) `

On a ` B = 7/(16)ln(3+ 2sqrt(2)) -4ln(1+sqrt(2)) -(25)/8ln(sqrt(2) -1) `

` = 7/(16)ln((sqrt(2)+1)^2 ) -4ln(1+sqrt(2)) -(25)/8ln(sqrt(2) -1) `

` = (7xx2)/(16)ln(sqrt(2)+1) -4ln(1+sqrt(2)) -(25)/8ln(sqrt(2) -1) `

` = (7/8 -4) ln(sqrt(2)+1) -(25)/8ln(sqrt(2) -1) `

` = ( -25)/8 ln(sqrt(2)+1) -(25)/8ln(sqrt(2) -1) `

` = -(25)/8[ ln(sqrt(2)+1) + ln(sqrt(2)-1) ] `

` = -(25)/8[ ln((sqrt(2)+1)(sqrt(2)-1)) ] `

` = -(25)/8[ ln(2-1) ] `

` = -(25)/8[ ln(1) ] `

` = 0 `

`=> B = 0 `

`D = 2ln(esqrt(e)) -3ln(e^3) -2ln(e^(-6)) `

On a `D = 2ln(esqrt(e)) -3ln(e^3) -2ln(e^(-6)) `

` = 2 ln(e) + 2ln(sqrt(e)) -3xx3 -2xx(-6) `

` = 2 + 2/2ln(e) -9 +12`

` = 3 +3 = 6 `

`=> D= 6 `

`E = ln(1/2) +ln(2/3) + ..... ln((98)/(99)) + ln((99)/(100)) `

On a `E = ln(1/2) +ln(2/3) + ..... ln((98)/(99)) + ln((99)/(100)) `

` = ln ( 1/2xx2/3 xx3/4 xx (98)/(99) xx (99)/(100)) `

` = ln(1/(100)) = -ln(10^2)= -2ln(10) `

`=> E = -2ln(10) `

`F = 3/8ln(7+4sqrt(3)) -ln(2+sqrt(3)) -1/4 ln(2-sqrt(3)) `

On a `F = 3/8ln(7+4sqrt(3)) -ln(2+sqrt(3)) -1/4 ln(2-sqrt(3)) `

` = 3/8ln( (2+sqrt(3))^2) -ln(2+sqrt(3)) -1/4 ln(2-sqrt(3)) `

` = 3/4 ln(2+sqrt(3)) -ln(2+sqrt(3)) -1/4 ln(2-sqrt(3)) `

` = (3/4 -1) ln(2+sqrt(3)) -1/4 ln(2-sqrt(3)) `

` = ( -1/4 ) ln(2+sqrt(3)) -1/4 ln(2-sqrt(3)) `

` = -1/4 [ ln(2+sqrt(3)) + ln(2-sqrt(3)) ] `

` = -1/4 [ ln(2^2 -sqrt(3)^2) ] `

` = -1/4 [ ln(4-3) ] `

` = -1/4 ln(1)= 0 `

`=> F = 0 `