Question

❌ $pàm.
$\frac{1}{3 \sqrt{2} { - 2 \sqrt{3} } }$
​

Answer 1

$\mathfrak{ Question : }$

$\frac{1}{3 \sqrt{2} { - 2 \sqrt{3} } }$

$\mathfrak{Points \: to \: Remember}$

$\mathsf{ \looparrowright How \: to \: rationalize \: the \: denominator}$

$\text{Refer to attachment for getting a proper idea}$

$\text{of rationalizing the denominator}$

Step 1: To rationalize the denominator, you must multiply both the numerator and the denominator by the conjugate of the denominator. Remember to find the conjugate all you have to do is change the sign between the two terms.

Step 2: Distribute (or FOIL) both the numerator and the denominator. Remember that you can multiply numbers outside the radical with numbers outside the radical and numbers inside the radical with numbers inside the radical.

Step 3: Combine like terms.

Step 4: Simplify the radicals.

Step 5: Combine like terms.

Step 6: Reduce the fraction, if you can. To reduce the fraction, you must reduce EACH number outside the radical by the same number. If you cannot reduce each number outside the radical by the same number, then the fraction cannot be reduced.

$\mathfrak{Solution : }$

$\frac{1}{3 \sqrt{2} { - 2 \sqrt{3} } } \: \times \frac{3 \sqrt{2} + 2 \sqrt{3} }{3 \sqrt{2} { + 2 \sqrt{3} } } \\ = \frac{3 \sqrt{2} + 2 \sqrt{3} }{(3 \sqrt{2})^{2} { - (2 \sqrt{3})^{2} } }$

$\frac{3 \sqrt{2} + 2 \sqrt{3} }{18 - 12 } = \frac{3 \sqrt{2} + 2 \sqrt{3} }{6 }$

$\text{《Refer to image for hand-written answer》}$

Answer 2

$\sf \bf \huge {\boxed {\mathbb {QUESTION}}}$

$\sf \bf \dfrac{1}{3 \sqrt{2} { - 2 \sqrt{3} } }$

$\sf \bf \huge {\boxed {\mathbb {ANSWER}}}$

$\sf \bf\implies \dfrac{1}{3 \sqrt{2} {- 2 \sqrt{3}}}$

${\underbrace {\overbrace {\color {orange} {\tt Multiply \:both \:numerator \:and\:denominator \:with\:conjugate\:of\:denominator \:by \:changing the\:sign}}}}$

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

$\sf \bf\implies \dfrac{1}{{\Big(3 \sqrt{2} {- 2 \sqrt{3}}\Big)}^{2}}$

$\sf \bf\implies \dfrac{3 \sqrt{2} + 2 \sqrt{3}}{{\Big[3 \sqrt{2} \Big]}^{2}- {\Big[2 \sqrt{3} \Big]}^{2}}$

$\sf \bf\implies \dfrac{3 \sqrt{2} + 2 \sqrt{3}}{{\Big[9 \big(2\big) \Big]}- {\Big[4 \big(3\big) \Big]}}$

$\sf \bf\implies \dfrac{3 \sqrt{2} + 2 \sqrt{3}}{18-12}$

$\implies{\boxed {\boxed {\color {blue} {\sf \bf \dfrac{3 \sqrt{2} + 2 \sqrt{3}}{6}}}}}$

${\color {purple}\underline {\tt Used\: Identity}}$

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

$\sf \bf \huge {\boxed {\mathbb {HOPE \:IT \:HELPS \:YOU }}}$

__________________________________________

$\sf \bf \huge {\boxed {\mathbb {EXTRA\:INFORMATION}}}$

${\color {green} {\tt Identities}}$

$\sf \bf {(a+b)}^{2}={a}^{2}+2ab+{b}^{2}$

$\sf \bf {(a-b)}^{2}={a}^{2}-2ab+{b}^{2}$

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