Find the value of a in (5 + 2√3) / (7 + 4√3) = a - 6√3

Solution:

Given, the expression is (5 + 2√3) / (7 + 4√3) = a - 6√3

We have to find the value of a.

Considering LHS,

LHS: (5 + 2√3) / (7 + 4√3)

By taking conjugate,

(5 + 2√3) / (7 + 4√3) = (5 + 2√3) / (7 + 4√3) × (7 - 4√3) / (7 - 4√3)

= (5 + 2√3)(7 - 4√3) / (7 + 4√3)(7 - 4√3)

By using algebraic identity,

(a² - b²) = (a - b)(a + b)

(7 + 4√3)(7 - 4√3) = (7)² - (4√3)²

= 49 - 16(3)

= 49 - 48

= 1

So, (5 + 2√3)(7 - 4√3) / (7 + 4√3)(7 - 4√3) = (5 + 2√3)(7 - 4√3) / (1)

= (5 + 2√3)(7 - 4√3)

By multiplicative and distributive property,

(5 + 2√3)(7 - 4√3) = 5(7) - 5(4√3) + 7(2√3) - 2√3(4√3)

= 35 - 20√3 + 14√3 - 8(3)

= 35 - 24 - 20√3 + 14√3

= 11 - 6√3

So, 11 - 6√3 = a - 6√3

By grouping,

11 - 6√3 + 6√3 = a

a = 11

Therefore, the value of a is 11.

✦ Try This: Find the values of a² + b² if a = 1/(7 - 4√3) and b = 1/(7 + 4√3)

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

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

Find the value of a in (5 + 2√3) / (7 + 4√3) = a - 6√3

Summary:

Rationalization can be considered as the process used to eliminate a radical or an imaginary number from the denominator of an algebraic fraction.The value of a in (5 + 2√3) / (7 + 4√3) = a - 6√3 is 11

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