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Find the square root of \[x = \sqrt {20.25} \].

Last updated date: 01st Mar 2024
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IVSAT 2024
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Here, we are required to find the square root of the given number. First of all, we will convert the decimal number into fraction. Then we will find the factors of the numerator of the obtained fraction using the prime factorization method. Then, we will consider a pair of the same numbers as a single number and multiply the remaining factors to find the required square root of the given number.

Complete Step by Step Solution:
We are given, \[x = \sqrt {20.25} \]
This can also be written as: \[x = \sqrt {\dfrac{{2025}}{{100}}} = \dfrac{{\sqrt {2025} }}{{\sqrt {100} }}\]………………………\[\left( 1 \right)\]
In this question, first we will do the prime factorization of 2025.
We can see that 2025 is an odd number, so dividing it by least odd prime number 3, we get
\[2025 \div 3 = 675\]
Now dividing 675 by 3, we get
\[675 \div 3 = 225\]
Now dividing 225 by 3, we get
\[225 \div 3 = 75\]
Now dividing 75 by 3, we get
\[75 \div 3 = 25\]
Now dividing 25 by 5, we get
\[25 \div 5 = 5\]
As we have obtained the quotient as a prime number, so we will not divide the number further.
Hence, 2025 can be written as:
\[2025 = 3 \times 3 \times 3 \times 3 \times 5 \times 5\]
Now, since we are required to find the square root, we will take only one prime number out of a pair of same prime numbers.
\[ \Rightarrow \sqrt {2025} = 3 \times 3 \times 5\]
Multiplying the terms, we get
\[ \Rightarrow \sqrt {2025} = 45\]
Hence, the square root of 2025 is 45.
Also, we already know that the square root of 100 is 10.
Because, 100 can be written as \[10 \times 10\]
Therefore from equation \[\left( 1 \right)\], we get
\[x = \dfrac{{\sqrt {2025} }}{{\sqrt {100} }} = \dfrac{{45}}{{10}} = 4.5\]

Therefore, the required answer is \[4.5\].

We know that factorization is a method of writing an original number as the product of its various factors. Also, prime numbers are those numbers which are greater than 1 and have only two factors, i.e. factor 1 and the prime number itself. Hence, prime factorization is a method in which we write the original number as the product of various prime numbers. In addition, the square root of a number is defined as the factors which when multiplied by itself gives the original number.
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