Thread: rationalising the denominator

Given that (2+square root of 7)(4-square root of 7) = a + b(square root of 7)
a) find the value of a and value of b

b) given that 2+(square of 7) / 4+(square of 7) = c +d(square of 7) where c and d are rational numbers,

find the value of c and value of d.

ummmm..

The first case involves multiplying out the brackets and then collecting like terms. You'll be able to collect terms with a factor of root 7.

The second case involves multipling the fraction by 1 expressed as ( 4 - root 7) / (4 - root 7). The point being that the denomitor can be multiplied out to remove the surd.

Given that (2+square root of 7)(4-square root of 7) = a + b(square root of 7)
a) find the value of a and value of b

[2 +sqrt(7)]*[4 -sqrt(7)] = a +b[sqrt(7)]
2*4 +2(-sqrt(7)) +sqrt(7)*4 +sqrt(7)*(-sqrt(7)) =
8 -2sqrt(7) +4sqrt(7) -7 =
1 +2sqrt(7) =
So, a=1 and b=2 ---------answer.

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b) given that 2+(square of 7) / 4+(square of 7) = c +d(square of 7) where c and d are rational numbers, find the value of c and value of d.

[2 +sqrt(7)] / [4 +sqrt(7)]
Rationalize the denominator. Multiply both numerator and denominator by the conjugant of the denominator, which is [4 -sqrt(7)],
= [2 +sqrt(7)]*[4 -sqrt(7)] / [4 +sqrt(7)]*[4 -sqrt(7)]
= [8 +2sqrt(7) -7] / [16 -7]
= [1 +2sqrt(7)] / 9
= (1/9) +(2/9)sqrt(7)
So, c = 1/9, and d = 2/9 ----------answer.