Question

What is the square root of -49?

Answer 1

Hint: For solving this type of question you should know about how to calculate the square root of any negative value. For calculating the square root of any negative value, we will get any imaginary value because it is impossible to get it as a real number. So, we always get an imaginary value for this.

Complete step by step solution:
Our question is asking to determine the square root of -49. And this is a negative value. As we know that there is no square which will give any negative value. Because if we calculate the square of any negative number then it will be multiplied by the same number with the same sign, so the sign will always be positive. But if we take about the imaginary numbers then it can be possible for a square to be a negative value. But it is impossible to be a negative square of any real value. So, if we see that, we know that ${{i}^{2}}=\left( -1 \right)$ and $i$ is an imaginary value which is known as iota.
So, for calculating the square root of any negative value or for calculating the square of any imaginary value, we use ${{\left( i \right)}^{2}}=-1$. So, according to our question,
${{x}^{2}}=-49$
Further we can write it as,
${{x}^{2}}=49{{\left( i \right)}^{2}}$
By taking the square root on both sides we get,
$\begin{align}
  & \sqrt{{{\left( x \right)}^{2}}}=\sqrt{49{{\left( i \right)}^{2}}} \\
 & \Rightarrow x=7\sqrt{{{\left( i \right)}^{2}}} \\
 & \Rightarrow x=7i \\
\end{align}$
Here the square root of -49 is equal to $7i$, which is an imaginary value that does not exist.

Note: During solving this type of question you should be careful with the negative and positive values which are given for calculating square root. And if the value is negative then it is always fixed that the value of that square root will be imaginary. And if the value is positive then the square root will be real.