What should come in place of both x in the equation	x	=	√162	.
	√128		x

L.C.M. of 21, 36, 66 = 2772.

Now, 2772 = 2 x 2 x 3 x 3 x 7 x 11

To make it a perfect square, it must be multiplied by 7 x 11.

So, required number = 2² x 3² x 7² x 11² = 213444

❨	√3 -	1	❩	2	= (√3)2 +	❨	1	❩	2	- 2 x √3 x	1
		√3					√3				√3

= 3 +	1	- 2
	3

= 1 +	1
	3

=	4
	3

(7 + 3√5)(7 - 3√5)

=

(7)2 - (3√5)2

= √49 - 45  = √4  = 2.

   2|64009( 253
    |4
    |----------
  45|240
    |225
    |----------
 503|  1509
    |  1509
    |----------
    |     X
    |---------- 

∴ √64009 = 253.

Let √0.0169 x x = 1.3.

Then, 0.0169x = (1.3)² = 1.69

⟹ x =	1.69	= 100
	0.0169

3√5 + √125 = 17.88

⟹ 3√5 + √25 x 5 = 17.88

⟹ 3√5 + 5√5 = 17.88

⟹ 8√5 = 17.88

⟹ √5 = 2.235

∴ √80 + 6√5 = √16 x 5 + 6√5

   = 4√5 + 6√5

   = 10√5 = (10 x 2.235) = 22.35

√4a² - 4a + 1 + 3a = √(1)² + (2a)² - 2 x 1 x 2a + 3a

   = √(1 - 2a)² + 3a

   = (1 - 2a) + 3a

   = (1 + a)

   = (1 + 0.1039)

   = 1.1039

   1|1.5625( 1.25
    |1
    |-------
  22| 56
    | 44
    |-------
 245| 1225
    | 1225
    |-------
    |    X
    |-------     

∴ √1.5625 = 1.25.

Let the three parts be A, B, C. Then,

A : B = 2 : 3 and B : C = 5 : 8 =	❨	5 x	3	❩	:	❨	8 x	3	❩	= 3 :	24
			5					5			5

⟹ A : B : C = 2 : 3 :	24	= 10 : 15 : 24
	5

⟹ B =	❨	98 x	15	❩	= 30.
			49

Originally, let the number of boys and girls in the college be 7x and 8x respectively.

Their increased number is (120% of 7x) and (110% of 8x).

⟹	❨	120	x 7x	❩	and	❨	110	x 8x	❩
		100					100

⟹	42x	and	44x
	5		5

∴ The required ratio =	❨	42x	:	44x	❩	= 21 : 22.
		5		5