Question

Find the square root of 256.

Answer 1

Hint: We solve this problem by using the prime factorisation method. The prime factorization method is the method of representing the numbers as the product of prime numbers.
So, we take the given number 256 and represent it as the product of prime numbers starting from the first prime number that is 2 so that we get the square of the product of some prime numbers then we can apply the square root so that we can evaluate the square root of 256 easily.

Complete step-by-step solution
We are asked to find the square root of 256.
Let us use the prime factorization method for the given number 256.
We know that the prime factorization method is the method of representing the numbers as the product of prime numbers
Now, let us take the first prime number that is 2
So, by expressing the number 256 as the product of first prime number 2 we get
\[\Rightarrow 256=2\times 128\]
Here we can see that the number 128 is not a prime number.
So, by expressing the number 128 as a product of prime number 2 we get
\[\Rightarrow 256={{2}^{2}}\times 64\]
Here we can see that the number 64 is not a prime number.
So, by expressing the number 64 as a product of prime number 2 we get
\[\Rightarrow 256={{2}^{3}}\times 32\]
Here we can see that the number 32 is not a prime number.
So, by expressing the number 32 as a product of prime number 2 we get
\[\Rightarrow 256={{2}^{4}}\times 16\]
Here we can see that the number 16 is not a prime number.
So, by expressing the number 16 as product of prime number 2 we get
\[\Rightarrow 256={{2}^{5}}\times 8\]
We know that the number 8 an be expressed as the power of 2 that is
\[\Rightarrow 8={{2}^{3}}\]
By using this result in above equation we get
\[\begin{align}
  & \Rightarrow 256={{2}^{5}}\times {{2}^{3}} \\
 & \Rightarrow 256={{2}^{8}} \\
\end{align}\]
Now, let us try to convert the RHS as the square of some number
We know that the standard formula of exponents that is
\[\Rightarrow {{x}^{a\times b}}={{\left( {{x}^{a}} \right)}^{b}}\]
By using this formula to above equation we get
\[\begin{align}
  & \Rightarrow 256={{2}^{4\times 2}} \\
 & \Rightarrow 256={{\left( {{2}^{4}} \right)}^{2}} \\
\end{align}\]
Now, by applying the square root on both sides we get
\[\begin{align}
  & \Rightarrow \sqrt{256}={{2}^{4}} \\
 & \Rightarrow \sqrt{256}=16 \\
\end{align}\]
Therefore we can conclude that the square root of 256 is 16.

Note: We can solve this problem in another method.
Here we can see that the number 256 has a unit digit of 4. So, we can say that the unit digit of the square root will be either 4 or 6.
Here, we can see that the number 256 is between 100 and 200 so that we can say that the square root of 256 lies between 10 and 20
So, by combining both the statements above we have two possibilities that are either 14 or 16 but,
\[\Rightarrow {{14}^{2}}=14\times 14=196\]
\[{{16}^{2}}=16\times 16=256\]
Therefore we can conclude that the square root of 256 is 16.