Question

Find the value of 2a - 1/√a + 1/a where a = 4+2√3​

Answer 1

`(19 + 6 √(3) )/(2)`

Explanation:

`2a - (1)/( √(a) ) + (1)/(a) \: \: \: when \: \: a = 4 + 2 √(3)`

`2 * (4 + 2 √(3) ) - \frac{1}{ \sqrt{4 + 2 √(3) } } + (1)/(4 + 2 √(3) )`

Expand the brackets (multiply every term inside the bracket by the term on the outside) and make the denominator the same (4 + 2√3):

`((8 + 4√(3) )*(4 + 2 √(3) ) )/(4 + 2 √(3) ) - \frac{ \sqrt{4 + 2 √(3) } }{4 + 2 √(3) } + (1)/(4 + 2 √(3) )`

`\frac{32 + 16 √(3) + 16 √(3) + 24 - \sqrt{4 + 2 √(3) } + 1}{4 + 2 √(3) }`

Collect like-terms:

`\frac{57+ 32 √(3) - \sqrt{4 + 2 √(3) } }{4 + 2 √(3) }`

`\frac{57 + 32 √(3) - \sqrt{( {1 + √(3)) }^(2) } }{4 + 2 √(3) }`

`(57 + 32 √(3) - (1 + √(3) ))/(4 + 2 √(3) )`

Eliminate the parentheses:

`(57 + 32 √(3) - 1 - √(3) )/(4 + 2 √(3) )`

Collect like terms:

`(56 + 31 √(3) )/(4 + 2 √(3) )`

`(56 + 31 √(3) )/(4 + 2 √(3) ) * (4 - 2 √(3) )/(4 - 2 √(3) ) = (224 - 112 √(3) + 124 √(3) - 186)/(16 - 4 * 3) = (38 + 12 √(3) )/(4)`

`(2(19 + 6 √(3)) )/(4) = (19 + 6 √(3) )/(2)`

Answer 2

Okay, let's substitute the value of a in the given expression:

2a - 1/√a + 1/a

= 2(4+2√3) - 1/√(4+2√3) + 1/(4+2√3)

= 8 + 4√3 - 1/(√(4+2√3)) + 1/(4+2√3)

Now let's simplify the expression by rationalizing the denominators:

= 8 + 4√3 - (1*(√(4+2√3)))/(√(4+2√3)*(√(4+2√3))) + (1*(√(4+2√3)))/(√(4+2√3)*(4+2√3))

= 8 + 4√3 - (√(4+2√3))/(4+2√3) + (√(4+2√3))/(4+2√3)

= 8 + 4√3

Therefore, the value of the expression is 8 + 4√3.