What is the square root of $405$? Explain it.

Question

Vedantu Content Team · Accepted Answer

Hint:  To find the square root of 405, we have to first find the factors of 405. We will do so using the prime factorisation method. After obtaining the factors, we will club the same factors in pairs of 2. If factors are not paired, we cannot find a square root. In that case, we use a long division method. We will follow the set of steps and find it.Complete step by step answer:Square root of a number is given by a number which when multiplied by itself yields the original number. The square root symbol is denoted by $\sqrt{{}}$ , also known as a radical symbol. Square root of a number can also be represented as the radical of a number, where the number inside the square root is called radicand.$\begin{align}  & y=\sqrt{x} \\  & \Rightarrow {{y}^{2}}=x \\ \end{align}$ Thus, x can also be called as the square of a number y.for a real valued input, square root of that input always gives a positive value.A perfect square is a number which can be prime factorized into even powers of its factors.Here, we need to find a square root of $405$ .To find the square root, we will need to figure out whether this number is a perfect square or not. If it is a perfect square, then its square root can be calculated using the prime factorization method. But, if it is not a perfect square, then we will have to use a long division method to find its square root.Prime factorization $405$ ,$\begin{align}  & 5\left| \!{\underline {\,  405 \,}} \right.  \\  & 9\left| \!{\underline {\,  81 \,}} \right.  \\  & 9\left| \!{\underline {\,  9 \,}} \right.  \\  & \text{  }\left| \!{\underline {\,  1 \,}} \right.  \\ \end{align}$ Thus, 405 can be expressed as $\begin{align}  & 405=5\times 9\times 9 \\  & \Rightarrow 405={{5}^{1}}\times {{9}^{2}} \\ \end{align}$ Since, all the factors of 405 can not be expressed in even powers, hence 405 is not a perfect square.So, we will use a long division method to find the square root of $405$ .We will start with 2 as a divisor and then we will write 2 in the quotient.Next, we will consider the divisor as twice the quotient, i.e. 4. Since 40 is the least possible divisor, we will move ahead with that and then add a decimal point.Next, we will consider double of 20, i.e. 40 in the divisor and now we have 500 as dividend. Considering 401 from possible options of 402, 403, … we will proceed with the division.We will do so till we obtain two decimal places in the quotient.$2\overset{20.12}{\overline{\left){\begin{align}  & 405.0000 \\  & \underline{4} \\  & 4\underline{0}\overline{\left){\begin{align}  & 05 \\  & \underline{00} \\  & 40\underline{1}\overline{\left){\begin{align}  & 500 \\  & \underline{401} \\  & 402\underline{2}\overline{\left){\begin{align}  & 9900 \\  & \underline{8044} \\  & 1856 \\ \end{align}}\right.} \\ \end{align}}\right.} \\ \end{align}}\right.} \\ \end{align}}\right.}}$ Hence, using a long division method up to two decimal places, we get a square root of $405$ as $20.12$.Note: To find the square root of a number, always prime factorize it first to determine whether it is a perfect square or not. This makes the process easier as applying a long division method is a lengthy and tedious task to do. While using a long division method to calculate the square root of a number, note that applying a single decimal point in the quotient would result in carrying two zeroes in the dividend. The same does not happen in simple division used to divide two numbers. So be careful in that.