Question

simplify by rationalizing the denominator root12+root18/root75-root50

Answer 1

Hey friend,
Here is the answer you were looking for:
$\frac{ \sqrt{12} + \sqrt{18} }{ \sqrt{75} - \sqrt{50} }$


We can split it and write it as

$\frac{ \sqrt{2 \times 2 \times 3} + \sqrt{2 \times 3 \times 3} }{ \sqrt{5 \times 5 \times 5} - \sqrt{5 \times 5 \times 2} } \\ \\ = \frac{ \sqrt{3 \times {2}^{2} } + \sqrt{2 \times {3}^{2} } }{ \sqrt{5 \times {5}^{2} } - \sqrt{2 \times {5}^{2} } } \\ \\ = \frac{2 \sqrt{3} + 3 \sqrt{2} }{5 \sqrt{5} - 5 \sqrt{2} }$

On rationalizing the denominator we get,

$\frac{2 \sqrt{3} + 3 \sqrt{2} }{5 \sqrt{5} - 5 \sqrt{2} } \times \frac{5 \sqrt{5} + 5 \sqrt{2} }{5 \sqrt{5} + 5 \sqrt{2} }$

Using the Identity:

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


$\frac{2 \sqrt{3} (5 \sqrt{5} + 5 \sqrt{2} ) + 3 \sqrt{2}(5 \sqrt{5} + 5 \sqrt{2} ) }{ {(5 \sqrt{5}) }^{2} - {(5 \sqrt{2}) }^{2} } \\ \\ = \frac{2 \sqrt{3} \times 5 \sqrt{5} +2 \sqrt{3} \times 5 \sqrt{2} + 3 \sqrt{2} \times 5 \sqrt{5} + 3 \sqrt{2} \times 5 \sqrt{2} }{125 - 50} \\ \\ = \frac{10 \sqrt{15} + 10 \sqrt{6} + 15 \sqrt{10} + 30}{75} \\ \\ = \frac{2 \sqrt{15} + 2 \sqrt{6} + 3 \sqrt{10} + 6 }{25}$

Hope this helps!!

If you have any doubt regarding to my answer, feel free to ask in the comment section or inbox me if needed.

@Mahak24

Thanks...
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