x = 4-√15
\(\frac { 1 }{ x } =\frac { 1 }{ 4-\sqrt { 15 } } \times \frac { 4+\sqrt { 15 } }{ 4+\sqrt { 15 } } \)
\(\frac { 1 }{ x } =\frac { 4+\sqrt { 15 } }{ 16-15 } \)
\(\frac { 1 }{ x } =4+\sqrt { 15 } \)
\({ \left( x+\frac { 1 }{ x } \right) }^{ 2 }={ \left( 4-\sqrt { 15 } +4+\sqrt { 15 } \right) }^{ 2 }\)
= (8)²
= 64

\(a=7-4\sqrt { 3 } =4+3-4\sqrt { 3 } \)
\(\\ ={ 2 }^{ 2 }+{ \left( \sqrt { 3 } \right) }^{ 2 }-2(2)(\sqrt { 3 } )\)
\(\\ ={ \left( 2-\sqrt { 3 } \right) }^{ 2 }\)
\(\\ \sqrt { a } =2-\sqrt { 3 } \quad ...(1)\)
\(\\ \frac { 1 }{ \sqrt { a } } =\frac { 1 }{ 2-\sqrt { 3 } }\)
\( \\ =\frac { 1 }{ 2-\sqrt { 3 } } \times \frac { 2+\sqrt { 3 } }{ 2+\sqrt { 3 } } =\frac { 2+\sqrt { 3 } }{ 4-3 } \)
\(\\ =2+\sqrt { 3 } \ ...(2)\)
From (1) and (2)
\(\sqrt { a } +\frac { 1 }{ \sqrt { a } } =\left( 2-\sqrt { 3 } \right) +\left( 2+\sqrt { 3 } \right) =4\)

\(=\frac { 40 }{ 2\sqrt { 10 } +\sqrt { 2\times 2\times 5 } +\sqrt { 2\times 2\times 10 } -2\sqrt { 5 } } \)
\(\\ =\frac { 40 }{ 2\sqrt { 10 } +\sqrt { 20 } +\sqrt { 40 } -2\sqrt { 5 } } \)
\(\\ =\frac { 40 }{ 4\sqrt { 10 } } =\frac { 10 }{ \sqrt { 10 } } \)
\(\\ \sqrt { 10 } =3.162\)

The two irrational numbers between 2 and 2.5 can be taken as
2.101001000100001...
2.201001000100001...

x = \(\frac{1}{7}\)

x = \(0.\overline{142857}\)

-0.00875 = \(\frac{875}{100000}\)
=\(\frac{35}{4000}=\frac{7}{800}\)

Draw a line XY and take a fixed length as unit length, which represents integers on this line.
(i) On the right side of O, take OA = 1 unit, then OB = 2 units. Divide the 3rd unit BC into 8 equal parts. Mark a point P on BC, such that BP represents 3/8 of a unit.
Therefore, OP represents \(2\frac { 3 }{ 8 } .\)
(ii) On the left side of O, take OD = 1 unit, then DE =1 unit. Divide the 2nd unit DE into 7 equal parts. Make a point Q on DE such that DQ represents 5/7 of a unit. 
Therefore, OQ represents \(-\left( 1\frac { 5 }{ 7 } \right) .\)

\(a=2,\quad b=\frac { 5 }{ 6 } \)

34

\(\frac { 4 }{ 3\sqrt { 3 } -2\sqrt { 2 } } +\frac { 3 }{ 3\sqrt { 3 } -2\sqrt { 2 } } =\frac { 21\sqrt { 3 } +2\sqrt { 2 } }{ 19 } \)
\(=\frac { 21(1.732)+2(1.414) }{ 19 } \)
\(=\frac { 39.2 }{ 19 } \)
= 2.063

\(\frac { 1 }{ 1+\sqrt { 2 } } +\frac { 1 }{ \sqrt { 2 } +\sqrt { 3 } } +\frac { 2 }{ \sqrt { 3 } +\sqrt { 5 } } \)
\(=\frac { 1 }{ \left( \sqrt { 2 } +1 \right) } \times \frac { \left( \sqrt { 2 } -1 \right) }{ \left( \sqrt { 2 } -1 \right) } +\frac { 1 }{ \left( \sqrt { 3 } +\sqrt { 2 } \right) } \times \frac { \left( \sqrt { 3 } -\sqrt { 2 } \right) }{ \left( \sqrt { 3 } -\sqrt { 2 } \right) } +\frac { 2 }{ \left( \sqrt { 5 } +\sqrt { 3 } \right) } \times \frac { \left( \sqrt { 5 } -\sqrt { 3 } \right) }{ \left( \sqrt { 5 } -\sqrt { 3 } \right) } \)
\(\frac { \left( \sqrt { 2 } -1 \right) }{ 2-1 } +\frac { \left( \sqrt { 3 } -\sqrt { 2 } \right) }{ 3-2 } +\frac { 2\left( \sqrt { 5 } -\sqrt { 3 } \right) }{ 5-3 } \)
= √2-1 + √3 - √2 + √5-√3 = √5 - 1

\(\frac { 214 }{ 33 } \)

\(\frac { 2+\sqrt { 3 } }{ 2-\sqrt { 3 } } =\frac { 2+\sqrt { 3 } }{ 2-\sqrt { 3 } } \times \frac { 2+\sqrt { 3 } }{ 2+\sqrt { 3 } } \)
\(=\frac { { (2+\sqrt { 3 } ) }^{ 2 } }{ { (2) }^{ 2 }-{ (\sqrt { 3 } ) }^{ 2 } } \left[ \because (a+b)(a-b)={ a }^{ 2 }-{ b }^{ 2 } \right] \)
\(=\frac { { (2+\sqrt { 3 } ) }^{ 2 } }{ 4-3 } \)
\(={ (2+\sqrt { 3 } ) }^{ 2 }=4+3+4\sqrt { 3 } \ \left[ \because \ { (a+b) }^{ 2 }={ a }^{ 2 }+{ b }^{ 2 }+2ab \right] \)
\(=7+4\sqrt { 3 } \)
\(But\quad \frac { 2+\sqrt { 3 } }{ 2-\sqrt { 3 } } =a+b\sqrt { 3 } \)
\(\because \ a+b\sqrt { 3 } =7+4\sqrt { 3 } \)
On comparing coefficient of a and b from both sides, we gieta=7, b=4
\(\therefore \ { a }^{ 2 }+{ b }^{ 2 }={ (7) }^{ 2 }+{ (4) }^{ 2 }=49+16=65\)

Let x = 0.0333    ...(i)
On multiplying Eq. (i) by 10, we get
10 x = 0.333    ....(ii)
Again, multiplying Eq. (ii) by 10, we get
100 x = 3.333     .....(iii)
On subtracting Eq. (ii) from Eq. (iii), we get
100 x - 10x = 3.333 ... - 0 .333 
\(\Rightarrow\) 90 x = 3
\(\therefore \ x=\frac { 3 }{ 90 } \)     ...(iv)
Again, let y = 0.444 ...      ...(v)
On multiplying Eq. (iv) by 10, we get
10y = 4.444 ...
On subtracting Eq. (iv) from Eq. (v), we get
10y - y = 4 .444 ... - 0.444 ...
\(\Rightarrow \ 9y=4\ \Rightarrow y=\frac { 4 }{ 9 } \)
\(\therefore\)Sum of 0.0333.... and 0.444...=x+y
\(=\frac { 3 }{ 90 } +\frac { 4 }{ 9 } =\frac { 3+40 }{ 90 } =\frac { 43 }{ 90 } \)