Can anyone help with the circled quesiton in the pic?  Thanks!

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Expert Answers

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a) You need to convert the expression `sqrt3(2 + sqrt3)` into the form `a + bsqrt3` , hence, you need to open the brackets performing the required the multiplications, such that:

`sqrt3(2 + sqrt3) = 2*sqrt3 + sqrt3*sqrt3`

`sqrt3(2 + sqrt3) = 2*sqrt3 + 3`

Re-arranging the terms, yields:

`sqrt3(2 + sqrt3) = 3 + 2*sqrt3`

Comapring the resulted form `3 + 2*sqrt3` to the requested form `a + bsqrt3` and equating the like parts, yields:

`a = 3`

`b = 2`

Hence, converting the expression `sqrt3(2 + sqrt3)` into the form `a + bsqrt3` yields `3 + 2*sqrt3` .

b) You need to convert the expression `4 - sqrt3 - 2(1 - sqrt3)` into the form `a + bsqrt3` such that:

`4 - sqrt3 - 2(1 - sqrt3) = 4 - sqrt3 - 2 + 2sqrt3`

Grouping the terms that contains `sqrt 3` , yields:

`4 - sqrt3 - 2(1 - sqrt3) = (4 - 2) + (2sqrt3 - sqrt3)`

`4 - sqrt3 - 2(1 - sqrt3) = 2 + sqrt3`

Comapring the resulted form `2 + sqrt3` to the requested form `a + bsqrt3` and equating the like parts, yields:

`a = 2`

`b = 1`

Hence, converting the expression `4 - sqrt3 - 2(1 - sqrt3)` into the form `a + bsqrt3` yields `2 + sqrt3` .

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