Question

Rationale the denominator and simplify 1+√2/3-2√2

Answer 1

$\frac{ 1 + \sqrt{2} }{3 - 2 \sqrt{2} } \\ = \frac{(1 + \sqrt{2} )(3 + 2 \sqrt{2} )}{(3 + 2 \sqrt{2} )(3 + 2 \sqrt{2} )} \\ = \frac{3 + 2 \sqrt{2} +3 \sqrt{2} + 4 }{ {(3)}^{2} + {(2 \sqrt{2} )}^{2} } \\ = \frac{ 7 + 5\sqrt{2} }{9 - 8} = 7+ 5\sqrt{2}$

Answer 2

$\huge\underline\mathfrak{Answer-}$

$\huge\leadsto$ $\huge\sf\red{7 + 5 \sqrt{2}}$

$\huge\underline\mathfrak{Explanation-}$

$\leadsto$ $\sf\dfrac{1 + \sqrt{2} }{3 - 2\sqrt{2} }$

By Rationalising the denominator,

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

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

By using,

★ (a+b)(a-b) = a² - b²

$\leadsto$ $\sf\dfrac{1(3 + 2 \sqrt{2} ) + \sqrt{2}(3 + 2 \sqrt{2} )}{ ({3})^{2} - ({2 \sqrt{2} })^{2} }$

$\leadsto$ $\sf\dfrac{3 + 2 \sqrt{2} + 3 \sqrt{2} + 2 \times 2 }{9 - (4 \times 2)}$

$\leadsto$ $\sf\dfrac{3 + 4 + 5 \sqrt{2} }{9 - 8}$

$\leadsto$ $\sf\dfrac{7 + 5 \sqrt{2} }{1}$

$\leadsto$ $\huge\sf\red{7 + 5 \sqrt{2}}$ ( required answer )