Question

if√2-√3/3root2+2 root3=a +b root6
where a and b rational numbers, then find the value of a and b
full explanation
it's urgent
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Answer 1

Answer:

Hence, the value of a = 2 and b = -5/6.

Step-by-step explanation:

Given:-

(√2-√3)/(3√2+2√3) = a+b√6

To find:-

Rationalised the denominator and find out the value of a and b.

Solution:-

Given that

(√2-√3)/(3√2+2√3) = a+b√6

Denominator = 3√2+2√3

We know that

Rationalising factor of a√b+a√b = a√b-a√b

Rationalising factor of 3√2+2√3 = 3√2-2√3

On Rationalising the denominator then

=> [(√2-√3)/(3√2+2√3)]×[(3√-2√3)/(3√2-2√3)]

=> [(√2-√3)(3√2-2√3)]/[(3√2+2√3)(3√2-2√3)]

Now, multiplying numerator left side to right side we get,

=> [3(√2×2)-2(√3×2)-3(√2×3)-2(√3×3)]/[(3√2+2√3)(3√2-2√3)]

=> (3×2-2√6-3√6-2×3)/([3√2+2√3)(3√2-2√3)]

=> (6-2√6-3√6-6)/[3√2+2√3)(3√2-2√3)]

(12-5√6)/[3√2+2√3)(3√2-2√3)]

Now, applying algebraic identity in denominator

We know that

(a+b)(a-b) = a^2-b^2

Where a = 3√2 and b = 2√3

=> (12-5√6)/[(3√2)^2-(2√3)^2]

=> (12-5√6)/(18-12)

=> (12-5√6)/6

The denominator 6 will cancel the numerator 12 in to times so the 12 will become 2.

=> 2 - ⅚√6

∴ a+b√6 = 2-⅚√6

On, comparing with R.H.S

a = 2,

b = -⅚√6 = -⅚

Answer:-

Hence, the value of a = 2 and b = -5/6.

Used formulae:-

Rationalising factor of a√b+√b = a√b-a√b

(a+b)(a-b)=a^2-b^2

:)