Question

please tell me all three questions fast

Answer 1

Hey dear,
Here is the answer you were looking for :
$(i) \frac{1}{2 - \sqrt{3} } - \frac{1}{ \sqrt{3} + \sqrt{2} } + \frac{5}{3 - \sqrt{2} }$

On rationalizing the denominators we get :

$\frac{1}{2 - \sqrt{3} } \times \frac{2 + \sqrt{3} }{2 + \sqrt{3} } - \frac{1 }{ \sqrt{3} + \sqrt{2} } \times \frac{ \sqrt{3} - \sqrt{2} }{ \sqrt{3} - \sqrt{2} } + \frac{5}{3 - \sqrt{2} } \times \frac{ 3 + \sqrt{2} }{3 + \sqrt{2} }$

Using the identity :

$(x + y)(x - y) = {x}^{2} - {y}^{2}$

$\frac{2 + \sqrt{3} }{ {(2)}^{2} - {( \sqrt{3}) }^{2} } - \frac{ \sqrt{3} - \sqrt{2} }{ {( \sqrt{3}) }^{2} - {( \sqrt{2}) }^{2} } + \frac{5(3 + \sqrt{2} )}{ {(3)}^{2} - {( \sqrt{2}) }^{2} } \\ \\ = \frac{2 + \sqrt{3} }{4 - 3} - \frac{ \sqrt{3} - \sqrt{2} }{3 - 2} + \frac{15 + 5 \sqrt{2} }{9 - 2} \\ \\ = 2 + \sqrt{3} - \sqrt{3} + \sqrt{2} + \frac{15 + 5 \sqrt{2} }{7} \\ \\ = 2 + \sqrt{2} + \frac{15 + 5 \sqrt{2} }{7} \\ \\ = \frac{2 \times 7 + \sqrt{2} \times 7 + 15 + 5 \sqrt{2} }{7} \\ \\ = \frac{14 + 7 \sqrt{2} + 15 + 5 \sqrt{2} }{7} \\ \\ = \frac{29 + 12 \sqrt{2} }{7}$

$3 .$

Let x be 3.8888...

x = 3.888.... (i)

Multiplying 10 to both the sides we get

10x = 38.888... (ii)

Subtracting both the equations :

10x - x = 38.888... - 3.888...

9x = 35

$x = \frac{35}{9}$
$4. \: (9( {64}^{ \frac{1}{3} } + {125}^{ \frac{1}{3} } )^{3} ) ^{ \frac{1}{4} } \\ \\ = (9( {4}^{3 \times \frac{1}{3} }+ {5}^{3 \times \frac{1}{3} } ) ^{3} )^{ \frac{1}{4} } \\ \\ = (9(4 + 5) ^{3} ) ^{ \frac{1}{4} } \\ \\ = ((36 + 45)^{3} )^{ \frac{1}{4} } \\ \\ = ( {(81)}^{3} ) ^{ \frac{1}{4} } \\ \\ = ( {531441})^{ \frac{1}{4} } \\ \\ = {27}^{4 \times \frac{1}{4} } \\ \\ = 27$

Hope this helps!!