Question

`(3sqrt(2) - 2sqrt(3))/(3sqrt(2) + 2sqrt(3))` after rationalisation becomes ______.

विकल्प

• `(6sqrt(2) - 4sqrt(3))/(6sqrt(2) + 4sqrt(3))`

• `9sqrt(2) - 4sqrt(3)`

• `1 + 2sqrt(6)`

• `5 - 2sqrt(6)`

Answer 1

`(3sqrt(2) - 2sqrt(3))/(3sqrt(2) + 2sqrt(3))` after rationalisation becomes `underlinebb(5 - 2sqrt(6))`.

Explanation:

Step 1: Multiply the numerator and denominator by the conjugate

To rationalize the denominator `3sqrt(2) + 2sqrt(3)`, we multiply both the numerator and the denominator by its conjugate, `3sqrt(2) - 2sqrt(3)`:

`(3sqrt(2) - 2sqrt(3))/(3sqrt(2) + 2sqrt(3)) xx (3sqrt(2) - 2sqrt(3))/(3sqrt(2) - 2sqrt(3))`

Step 2: Expand the numerator and denominator

We expand the terms using the identities (a – b)² = a² – 2ab + b² for the numerator and (a + b)(a – b) = a² – b² for the denominator:

Numerator:

`(3sqrt(2) - 2sqrt(3))^2`

= `(3sqrt(2))^2 - 2(3sqrt(2))(2sqrt(3)) + (2sqrt(3))^2`

= `(9 xx 2) - 12sqrt(16) + 12`

= `18 - 12sqrt(6) + 12`

= `30 - 12sqrt(6)`

Denominator:

`(3sqrt(2) + 2sqrt(3))(3sqrt(2) - 2sqrt(3))`

= `(3sqrt(2))^2 - (2sqrt(3))^2`

= (9 × 2) – (4 × 3)

= 18 – 12

= 6

Step 3: Simplify the resulting fraction

Now we combine the simplified numerator and denominator into a single fraction:

`(30 - 12sqrt(6))/6`

We can factor out the common factor of 6 from the numerator:

`(6(5 - 2sqrt(6)))/6`

Cancel the 6 from the numerator and denominator:

`5 - 2sqrt(6)`