Resolving 256 into prime factors, 256 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2
Grouping the factors in pairs in such a way that both the factors in each pair are same.
256 = 2² x 2² x 2² x 2²
Here, 256 can be grouped into pairs of equal factors and no factor is left over.

Hence, 256 is a square number.
Again, 2 X 2 x 2 X 2 = 16
50,256 is the square of 16.

According to Pythagoras theorem,
(Hypotenuse)² = (Base)² + (Perpendicular)²
Clearly, the side opposite to 90° is hypotenuse
AC = Hypotenuse
AB = Base
BC = Perpendicular

(AC)²  = (AB)² + (BC)²
\(\Rightarrow\) (AC)² =(12)² +(5)²
\(\Rightarrow\) (AC)²=144+25
\(\Rightarrow\) (AC)² =169
\(\Rightarrow\) AC =\(\sqrt { 169 } \) = 13 cm

Here, 324 -1 = 323,323 - 3 = 320
320 - 5 = 315,315 -7 = 308
308 - 9 = 299, 299 - 11 = 288
288 -13 = 275,275 -15 = 260
260 - 17 = 243, 243 - 19 = 224
224 - 21 = 203,203 - 23 =180
180 - 25 =155, 155 - 27 =128
128 - 29 = 99, 99 - 31 = 68
68 - 33 = 35, 35 - 35 = 0
So, to get 0, we use 18 steps.
Hence, square root of 324 is 18.

We have, 1500

We see that, 38² < 1500 < 39²
So, number to be added = 39² - 1500
= 1521 - 1500 = 21
Therefore, the perfect square is 1500 + 21 = 1521 and \(\sqrt { 1521 } \)= 39
So, the required number is 21 and the square root is 39.

Let AC be the ladder. Therefore.
AC=10 m

Let BC be the distance between the foot of the ladder and the wall.
Therefore,BC = 6 m
\(\Delta\) ABC is a right angled triangle, right angled at B.
By using Pythagoras theorem,
AC² = AB² + BC2
\(\Rightarrow\) 10² = AB² + 6²
\(\Rightarrow\) AB² =10² - 6²
\(\Rightarrow\) AB² = 100 - 36 = 64
\(\therefore\) AB = \(\sqrt { 64 } \) = 8m
Hence, the wall is 8 m high.

To find the square roots of 100, we subtract successive odd number starting from 1 as follows:
100 - 1 = 99, 99 - 3 = 96, 96 - 5 = 91, 91 - 7 = 84
84 -9 = 75, 75 -11 =64, 64 -13 = 51,51-15 =36
36-17 = 19, 19 - 19 = 0
We observe that the number 10 reduced to zero after subtracting first 10 odd numbers.
So, 100 is a perfect square.
\(\therefore \ \sqrt { 100 } =10\)
Hence, the square root of 100 is 10.

16

1

Given that, \(\sqrt { 248+\sqrt { 52+\sqrt { 144 } } } =\sqrt { 248+\sqrt { 52+12 } } \)
\(\left[ \because 144={ (12) }^{ 2 } \right] \)
\(=\sqrt { 248+\sqrt { 64 } } =\sqrt { 248+8 } [\because 64={ (8) }^{ 2 }]\)
\(=\sqrt { 256 } =16\)
Hence, value of the expression is 16.

\(\therefore \ \sqrt { 5625 } =75\)