Question

find the values of a and b in each of the following 5 + 2 root 3 by 7 + 4 root 3 is equal to a + b root 3​

Answer 1

a=11

b=-6

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Answer 2

$Solutions:-$

• $\frac{5 + 2\sqrt{3}}{7 + 4 \sqrt{3}} = a + b\sqrt{3}$

$To \: Find:-$

• $Value \: \: of \: \: a \: \: and \: \: b$

$Solutions:-$

• $\frac{5 + 2\sqrt{3}}{7 + 4 \sqrt{3}} = a + b\sqrt{3}$

$Taking \: \: \: LHS:-$

$⇢ \frac{5 + 2 \sqrt{3}}{7 + 4 \sqrt{3}}$

$Rationalising \: \: \: denominator:-$

$⇢ \frac{5 + 2 \sqrt{3}}{7 + 4 \sqrt{3}} × \frac{7 - 4 \sqrt{3}}{7 - 4 \sqrt{3}}$

$(a + b)(a - b) = {a}^{2} - {b}^{2}$

$⇢ \frac{5(7 + 4\sqrt{3} + 2\sqrt{3}(7 + 4\sqrt{3})}{49 - 48}$

$⇢ \frac{35 - 20\sqrt{3} + 14\sqrt{3} - 24}{1}$

$It \: \: is \: \: given \: \: that \: LHS = a + b\sqrt{3}$

$⇢ \frac{11 - b\sqrt{3}}{1} = a + b\sqrt{3}$

$⇢ 11 = a$

$⇢ b\sqrt{3} = -6\sqrt{3}$

$⇢ b = -6$

$Hence, \: \: a = 11 \: \: \: or \: \: \: b = -6$

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