Question

What is the square root of $ - 340$?

Answer 1

Hint: Square root of a number is a value, which when multiplied by itself gives the original number. Suppose, ‘x’ is the square root of ‘y’, then it is represented as $x = \sqrt y $ or we can express the same equation as ${x^2} = y$. In the given question, we are required to calculate the root of a negative number provided to us in the problem. Here we can see that $ - 340$ is not a perfect square. Now, to simplify the square root, we first do the prime factorization of the number $340$ and take the factors occurring in pairs outside of the square root radical.

Complete step by step answer:
We are required to evaluate the square root of $\left( { - 340} \right)$.
$340$ can be factorized as,
$340 = 2 \times 2 \times 5 \times 17$
Now, expressing the prime factorization in powers and exponents, we get,
$340 = {2^2} \times 5 \times 17$
We can see that $2$ is multiplied twice and hence the power of $2$ is two.
Now, we find the square root of $\left( { - 340} \right)$ as,
Now, \[\sqrt { - 340} = \sqrt {\left( { - 1} \right) \times {2^2} \times 5 \times 7} \]

Since we know that ${2^2}$ is a perfect square. So, we can take this outside of the square root we have,
\[ \Rightarrow \sqrt { - 340} = 2\sqrt {\left( { - 1} \right) \times 5 \times 7} \]
Since $7$ and $5$ are not perfect squares, we can multiply this and keep it inside the square root,
\[ \Rightarrow \sqrt { - 340} = 2\sqrt {\left( { - 1} \right) \times 35} \]
Now, we know that the square root of $\left( { - 1} \right)$ is $i$. So, we get,
\[ \therefore \sqrt { - 340} = 2\sqrt {35} i\]

Hence, the square root of $ - 340$ is \[2\sqrt {35} i\].

Note: Here $\sqrt {} $ is the radical symbol used to represent the root of numbers.The number under the radical symbol is called radicand. The positive number, when multiplied by itself, represents the square of the number. The square root of the square of a positive number gives the original number. Now, we must know the square root of $\left( { - 1} \right)$ to solve such questions.