Question

rationalize the denominator​

Answer 1

1/(5 - 2√3)

1/(5 - 2√3) * (5 + 2√3)/(5 + 2√3)

(5 + 2√3)/((5)² - (2√3)²)

(5 + 2√3)/(25 - 12)

(5 + 2√3)/13

Answer 2

Question :-

$\sf Rationalize \:\: \dfrac{1}{5-2\sqrt{3}}$

Answer :-

In order to rationalise the denominator, we have to multiply the numerator and the denominator but the denominator's inverse such that (a-b)(a+b) = a² - b² is formed in the denominator.

• Denominator = 5-2√3
• Rationalising factor = 5 + 2√3

$\sf Multiplying\:by\:\: \dfrac{5+\sqrt{3}}{5+\sqrt{3}}$

$\implies \sf \dfrac{1}{5 - 2\sqrt{3} } \times \dfrac{5 + 2\sqrt{3} }{5 + 2\sqrt{3} }$

$\implies \sf \dfrac{1(5 + 2\sqrt{3}) }{(5 - 2\sqrt{3})(5 + 2\sqrt{3}) }$

$\implies \sf \dfrac{5 + 2\sqrt{3} }{ {(5)}^{2} - (2 \sqrt{3})^{2} }$

$\implies \sf \dfrac{5 + 2 \sqrt{3} }{(5 \times 5) - (2 \times 2 \times \sqrt{3} \times \sqrt{3} )}$

$\implies \sf \dfrac{5 + 2 \sqrt{3} }{(25) - (12)}$

$\implies \sf \dfrac{5 + 2 \sqrt{3} }{13}$

$Therefore,\: the\: Rationalized\: form\: of \:\:\: \dfrac{1}{5-2\sqrt{3}} \:\:\: is \:\:\: \dfrac{5+\sqrt{3}}{13}$

Identities :-

• (a+b)² = a² + 2ab + b²
• (a-b)² = a² - 2ab + b²
• a² - b² = (a+b)(a-b)
• (x+a)(x+b) = x² + x(a+b) + ab
• (a+b+c)² = a² + b² + c² + 2(ab + bc + ca)
• (a+b)³ = a³ + 3ab(a+b) + b³
• (a-b)³ = a³ - 3ab(a-b) - b³
• a³ + b³ = (a+b)(a² - ab + b²)
• a³ - b³ = (a-b)(a² + ab + b²)