Question

What is the square root of 60?

Answer 1

Hint: For solving this question you should know about the square root of any whole number. Square root of any whole number is equal to $\dfrac{1}{2}$ power of that number or if we take under root of any number then we get the square root of that number or we can also find the square root by doing factors and then taking these common of root and solve that.

Complete step-by-step solution:
According to our question we have to solve the square root of 60. As we know that the square root of any number is equal to $\dfrac{1}{2}$ power of that number. So, if we calculate the square root of 60, then consider that the number is $x$ and the square of that is 60. So, we can write,
${{x}^{2}}=60$
Now take the under root on both sides, so we get,
$\sqrt{{{x}^{2}}}=\sqrt{60}$
Or we can also write it as,
$\begin{align}
  & {{\left( {{x}^{2}} \right)}^{\dfrac{1}{2}}}={{\left( 60 \right)}^{\dfrac{1}{2}}} \\
 & \Rightarrow x={{\left( 60 \right)}^{\dfrac{1}{2}}} \\
 & \Rightarrow x={{\left( 2\times 2\times 3\times 5 \right)}^{\dfrac{1}{2}}} \\
 & \Rightarrow x={{\left( {{2}^{2}}\times 3\times 5 \right)}^{\dfrac{1}{2}}} \\
\end{align}$
If we solve for $x$, then,
$\begin{align}
  & x=2{{\left( 3\times 5 \right)}^{\dfrac{1}{5}}} \\
 & \Rightarrow x=2{{\left( 15 \right)}^{\dfrac{1}{5}}} \\
 & \Rightarrow x=2\sqrt{15} \\
\end{align}$
Now as we know that $\sqrt{15}$ will not simplify further, but we can find rational approximations for it using Newton Raphson type method. So, according to this method,
Let $n=15,{{p}_{o}}=4,{{q}_{0}}=1$ and iterate using the formula,
$\begin{align}
  & {{p}_{i}}+1={{p}_{i}}^{2}+n{{q}_{i}}^{2} \\
 & {{q}_{i}}+1=2{{p}_{i}}{{q}_{i}} \\
\end{align}$
And at each iteration, $\dfrac{{{p}_{i}}}{{{q}_{i}}}$ is a rational approximation for $\sqrt{n}$ . So,
$\begin{align}
  & {{p}_{1}}={{p}_{0}}^{2}+n{{q}_{0}}^{2}={{4}^{2}}+{{15.1}^{2}}=16+15=31 \\
 & {{q}_{1}}=2{{p}_{0}}{{q}_{0}}=2.4.1=8 \\
\end{align}$
Then,
$\begin{align}
  & {{p}_{2}}={{p}_{1}}^{2}+n{{q}_{1}}^{2}={{\left( 31 \right)}^{2}}+{{15.8}^{2}}=961+960=1291 \\
 & {{q}_{2}}=2{{p}_{1}}{{q}_{1}}=2.31.8=496 \\
\end{align}$
So, we can write $\sqrt{15}=\dfrac{1290}{496}$
So,
$\sqrt{60}=2\sqrt{15}=2.\dfrac{1291}{496}=\dfrac{1291}{248}$
Therefore, the square root of 60 is $2\sqrt{15}$ or $\dfrac{1291}{248}$.

Note: During calculating the square root of any number we always have to be careful because if there is any calculation mistake then the complete solution would be wrong. And here only the calculations are the necessary step. And for calculating the full solution of the root, use Newton Raphson method.