Find the value of b in (√2 + √3) / (3√2 - 2√3) = 2 - b√6

Solution:

Given, the expression is (√2 + √3) / (3√2 - 2√3) = 2 - b√6

We have to find the value of b

Considering LHS,

LHS: (√2 + √3) / (3√2 - 2√3)

By taking conjugate,

(√2 + √3) / (3√2 - 2√3) = (√2 + √3) / (3√2 - 2√3) × (3√2 + 2√3) / (3√2 + 2√3)

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

(a² - b²) = (a - b)(a + b)

(3√2 - 2√3)(3√2 + 2√3) = (3√2)² - (2√3)²

= 9(2) - 4(3)

= 18 - 12

= 6

So, (√2 + √3)(3√2 + 2√3) / (3√2 - 2√3)(3√2 + 2√3) = (√2 + √3)(3√2 + 2√3) / 6

(√2 + √3)(3√2 + 2√3) = √2(3√2) + √2(2√3) + √3(3√2) + √3(2√3)

= 3(2) + 2√6 + 3√6 + 2(3)

= 6 + 5√6 + 6

= 12 + 5√6

Now, (√2 + √3)(3√2 + 2√3) / 6 = (12 + 5√6) / 6

= 12/6 + 5√6/6

= 2 + 5√6/6

Now, 2 - b√6 = 2 + 5√6/6

2 - b√6 - 2 = 5√6/6

-b√6 = 5√6/6

-b = 5/6

Therefore, b = -5/6

✦ Try This: Find the values of a³ + b³ if a = 1/(3 - 2√3) and b = 1/(3 + 2√3)

☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 1

NCERT Exemplar Class 9 Maths Exercise 1.3 Problem 11(iii)

Summary:

A conjugate is a similar surd but with a different sign. The value of b in (√2 + √3) / (3√2 - 2√3) = 2 - b√6 is -5/6

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