Question

2√5+3√5/2√5-3√2, Rationalize the given Denominator.​

Answer 1

Answer:

2√5+3√5/2√5-3√2

11.972136-3√2

7.72949531

Answer 2

●The rationalising of the denominator of the given fraction = $\bf\dfrac{50+15\sqrt{10}}{2}$

$\large\bold{\underline{\underline{Given:-}}}$

A fraction, $\bf \dfrac{2 \sqrt{5} + 3 \sqrt{5}}{2 \sqrt{5} - 3 \sqrt{2}}$

$\large\bold{\underline{\underline{Required:-}}}$

$\bf \dfrac{2 \sqrt{5} + 3 \sqrt{5}}{2 \sqrt{5} - 3 \sqrt{2}}$, Rationalize the Denominator for given Fraction.

$\large\bold{\underline{\underline{Solution:-}}}$

Given fraction,

$\bullet \: \: \: \: \bf \dfrac{2 \sqrt{5} + 3 \sqrt{5}}{2 \sqrt{5} - 3 \sqrt{2}}$

Multiple the reciprocal of the denominator by both the denominator and the numerator of the given fraction.

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

Using identity in the denominator of the given fraction.

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

$\\ \bf \implies \dfrac{(2 \sqrt{5} + 3 \sqrt{5} )(2 \sqrt{5} + 3 \sqrt{2})}{ {(2 \sqrt{5} )}^{2} - {(3 \sqrt{2})}^{2} } \\ \\ \\ \bf \implies \dfrac{2 \sqrt{5}(2 \sqrt{5} + 3 \sqrt{2}) + 3 \sqrt{5}(2 \sqrt{5} + 3 \sqrt{2})}{ {(2 \sqrt{5})}^{2} - {(3 \sqrt{2})}^{2} } \\ \\ \\ \bf \implies \dfrac{ {(2 \sqrt{5}) }^{2} + (2 \times 3 \sqrt{5 \times 2} ) + (3 \times 2 \sqrt{5 \times 5} + (3 \times 3 \sqrt{5 \times 2})}{4 \times 5 - 9 \times 2} \\ \\ \\ \bf \implies \dfrac{4 \times 5 + 6 \sqrt{10} + 6 \times 5 + 9 \sqrt{10}}{20 - 18} \\ \\ \\ \bf \implies \dfrac{20 + 6 \sqrt{10} + 30 + 9 \sqrt{10}}{2} \\ \\ \\ \bf \implies \dfrac{20 + 30 + 6 \sqrt{10} + 9 \sqrt{10} }{2} \\ \\ \\ \bf \implies \dfrac{50 + 15 \sqrt{10} }{2}$

Hence,

The rationalising of the denominator of the given fraction is $\bf\dfrac{50+15\sqrt{10}}{2}$