Question

Using the square root table, find the square root of 25725.

Answer 1

Hint: Recall that $ {{16}^{2}}=256 $ , or $ {{160}^{2}}=25600 $ , which is quite close to 25725.
Factorize the given number into its prime factors and group them into even powers.
Look up the square root table for square roots of leftover single prime factors.

Complete step-by-step answer:
The given number 25725 is a multiple of 25 (last two digits are a multiple of 25). It can be factorized as follows:
 $ 25725=1029\times 25 $
Now, 1029 is divisible by 3 because the sum of its digits $ (1+0+2+9=12=1+2=3) $ is a multiple of 3.
⇒ $ 25725=3\times 343\times 25 $
Recall that $ 343={{7}^{3}}$ .
∴ $ 25725=3\times {{5}^{2}}\times {{7}^{3}}$
Separating the even powers, we get:
⇒ $ 25725=3\times 7\times {{5}^{2}}\times {{7}^{2}}$
Taking the square root, we get:
⇒ $\sqrt{25725}=\sqrt{3}\times \sqrt{7}\times 5\times 7 $
From the table of square roots, we see that $\sqrt{3}=1.732 $ and $\sqrt{7}=2.645 $ .
⇒ $\sqrt{25725}=1.732\times 2.645\times 5\times 7 $
On multiplying the numbers together, we get:
⇒ $\sqrt{25725}=160.3399 $

Note: Square roots of prime numbers from 1 to 100: (total 25 primes). Remembering the given table will help us to solve it easily.

$\sqrt{2}$	1.4142	$\sqrt{13}$	3.6056	$\sqrt{31}$	5.5678	$\sqrt{53}$	7.2801	$\sqrt{73}$	8.5440
$\sqrt{3}$	1.7321	$\sqrt{17}$	4.1231	$\sqrt{37}$	6.0828	$\sqrt{59}$	7.6811	$\sqrt{79}$	8.8882
$\sqrt{5}$	2.2361	$\sqrt{19}$	4.3589	$\sqrt{41}$	6.4031	$\sqrt{61}$	7.8102	$\sqrt{83}$	9.1104
$\sqrt{7}$	2.6458	$\sqrt{23}$	4.7958	$\sqrt{43}$	6.5574	$\sqrt{67}$	8.1854	$\sqrt{89}$	9.4340
$\sqrt{11}$	3.3166	$\sqrt{29}$	5.3852	$\sqrt{47}$	6.8557	$\sqrt{71}$	8.4261	$\sqrt{97}$	9.8489