Question

Find the value of $\sqrt {15625} $ and evaluate $\sqrt {156.25} + \sqrt {1.5625} $

Answer 1

Hint: For finding the square root we use prime factorization method. In this method we factorize the numbers into prime factors and then evaluate the given expression.

Complete step-by-step answer:
The objective of the problem is to find the value of $\sqrt {15625} $ and evaluate $\sqrt {156.25} + \sqrt {1.5625} $
To find the value of $\sqrt {15625} $ we use a long division method of prime factorization.
For this first we divide the number by the smallest prime number which is exactly divisible.
Again we divide the obtained quotient by the same or another smallest prime number. Repeat the process until quotient becomes one.
Now consider ,
$
  5\left| \!{\underline {\,
  {15625} \,}} \right. \\
  5\left| \!{\underline {\,
  {3125} \,}} \right. \\
  5\left| \!{\underline {\,
  {625} \,}} \right. \\
  5\left| \!{\underline {\,
  {125} \,}} \right. \\
  5\left| \!{\underline {\,
  {25} \,}} \right. \\
  5\left| \!{\underline {\,
  5 \,}} \right. \\
  \,\,\,1 \\
$
Now we express the given number into a product of prime numbers.
$15625 = 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 1$
Now take the square roots on both sides we get $\sqrt {15625} = \sqrt {5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 1} $
$\sqrt {15625} = 5 \times 5 \times 5$
Now multiply the factors to get the square root that is the square root of 15625 is 125.
Hence the square root of 15625 is a perfect square.
Now we have to evaluate the value of $\sqrt {156.25} + \sqrt {1.5625} $
Let us consider $\sqrt {156.25} + \sqrt {1.5625} $
Dividing the first term with 100 and second term with 1000 because to make the numbers decimal free. That is
$ = \sqrt {\dfrac{{15625}}{{100}}} + \sqrt {\dfrac{{15625}}{{1000}}} $
Now we using the formula $\sqrt {\dfrac{a}{b}} = \dfrac{{\sqrt a }}{{\sqrt b }}$
$ = \dfrac{{\sqrt {15625} }}{{\sqrt {100} }} + \dfrac{{\sqrt {15625} }}{{\sqrt {10000} }}$
Now simplifying the above equation , we get
$
   = \dfrac{{125}}{{10}} + \dfrac{{125}}{{100}} \\
   = 12.5 + 1.25 \\
   = 13.75 \\
$
Thus , the value of $\sqrt {156.25} + \sqrt {1.5625} $ is 13.75.

Note: The prime factors of a number are the prime numbers that when multiplied it gives the original number. There are two methods of prime factorization: the division method and the factor tree method. The first method is already described in the above step by step solution and the second method factor tree is used to determine factors of natural numbers which are greater than one.