#### Give a and b are rational numbers. Find a and b if:

(i) [3 – √5] / [3 + 2√5] = -19/11 + a√5

(ii) [√2 + √3] / [3√2 – 2√3] = a – b√6

(iii) {[7 + √5]/[7 – √5]} – {[7 – √5]/[7 + √5]} = a + 7/11 b√5

**Solution:**

(i) [3 – √5] / [3 + 2√5] = -19/11 + a√5

Let us consider LHS

[3 – √5] / [3 + 2√5]

Rationalize the denominator,

[3 – √5] / [3 + 2√5] = [(3 – √5) (3 – 2√5)] / [(3 + 2√5) (3 – 2√5)]

= [3(3 – 2√5) – √5(3 – 2√5)] / [3^{2} – (2√5)^{2}]

= [9 – 6√5 – 3√5 + 2.5] / [9 – 4.5]

= [9 – 6√5 – 3√5 + 10] / [9 – 20]

= [19 – 9√5] / -11

= -19/11 + 9√5/11

**So when comparing with RHS**

-19/11 + 9√5/11 = -19/11 + a√5

Hence, value of a = 9/11

(ii) [√2 + √3] / [3√2 – 2√3] = a – b√6

Let us consider LHS

[√2 + √3] / [3√2 – 2√3]

Rationalize the denominator,

[√2 + √3] / [3√2 – 2√3] = [(√2 + √3) (3√2 + 2√3)] / [(3√2 – 2√3) (3√2 + 2√3)]

= [√2(3√2 + 2√3) + √3(3√2 + 2√3)] / [(3√2)^{2} – (2√3)^{2}]

= [3.2 + 2√2√3 + 3√2√3 + 2.3] / [9.2 – 4.3]

= [6 + 2√6 + 3√6 + 6] / [18 – 12]

= [12 + 5√6] / 6

= 12/6 + 5√6/6

= 2 + 5√6/6

= 2 – (-5√6/6)

**So when comparing with RHS**

2 – (-5√6/6) = a – b√6

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

(iii) {[7 + √5]/[7 – √5]} – {[7 – √5]/[7 + √5]} = a + 7/11 b√5

Let us consider LHS

Since there are two terms, let us solve individually

{[7 + √5]/[7 – √5]}

Rationalize the denominator,

[7 + √5]/[7 – √5] = [(7 + √5) (7 + √5)] / [(7 – √5) (7 + √5)]

= [(7 + √5)^{2}] / [7^{2} – (√5)^{2}]

= [7^{2} + (√5)^{2} + 2.7.√5] / [49 – 5]

= [49 + 5 + 14√5] / [44]

= [54 + 14√5] / 44

Now,

{[7 – √5]/[7 + √5]}

Rationalize the denominator,

[7 – √5]/[7 + √5] = (7 – √5) (7 – √5)] / [(7 + √5) (7 – √5)]

= [(7 – √5)^{2}] / [7^{2} – (√5)^{2}]

= [7^{2} + (√5)^{2} – 2.7.√5] / [49 – 5]

= [49 + 5 – 14√5] / [44]

= [54 – 14√5] / 44

So, according to the question

{[7 + √5]/[7 – √5]} – {[7 – √5]/[7 + √5]}

By substituting the obtained values,

= {[54 + 14√5] / 44} – {[54 – 14√5] / 44}

= [54 + 14√5 – 54 + 14√5]/44

= 28√5/44

= 7√5/11

**So when comparing with RHS**

7√5/11 = a + 7/11 b√5

Hence, value of a = 0 and b = 1

**More Solution:**

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