Answer

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**Hint:**

Here, we have to find the square root of a given number. We can find the square root of the number by using the Prime Factorization methods. The square root of any number is equal to a number, which when squared gives the original number.

Formula Used: The square root of a number is given by the formula: \[y = \sqrt a \]where \[a\] is the given number.

**Complete step by step solution:**

We have to find the square root of 256.

Now, we are using the prime factorization method to find the square root.

Now, we have to find the prime factors.

As 256 is an even number, we will start dividing it by least prime number 2. Therefore, we get

\[256 \div 2 = 128\]

Now 128 is again an even number, so dividing it by 2, we get

\[128 \div 2 = 64\]

64 is again an even number, so dividing it by 2, we get

\[64 \div 2 = 32\]

Now dividing 32 by 2, we get

\[32 \div 2 = 16\]

Dividing 16 by 2, we get

\[16 \div 2 = 8\]

Dividing 16 by 2, we get

\[8 \div 2 = 4\]

Dividing 16 by 2, we get

\[4 \div 2 = 2\]

Now as we have got the quotient as a prime number, we will not divide the number further.

Thus, we can write 256 as

\[256 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\]

Now, we have to find the square root by using the prime factors.

Taking square root on both sides of the above equation, we get

\[ \Rightarrow \sqrt {256} = \sqrt {2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2} \]

Now, we will get

\[ \Rightarrow \sqrt {256} = 2 \times 2 \times 2 \times 2\]

Multiplying the terms, we have

\[ \Rightarrow \sqrt {256} = 16\]

**Therefore, the square root of the number 256 is 16.**

**Note:**

The square root of the number can also be found using the method of long division method. The Long Division method is a method which can be used for imperfect numbers too.

Therefore, the square root of 256 is 16.

If a number ends with 2, 3, 7 or 8 (in the unit digit), then the perfect square root does not exist. If a number ends with 1, 4, 5, 6 or 9 in the unit digit, then the number will have a square root. Prime factorization is a process of factoring a number in terms of prime numbers i.e. the factors will be prime numbers.

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