Question

What is the square root of -16?

Answer 1

Hint: The square root of -1 is expressed as \[i\]. To solve the given question, we should know some of the algebraic properties. We should know that \[\sqrt{a}\] can also be written as \[{{a}^{\dfrac{1}{2}}}\]. The other exponential property we should know is \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}\], also \[{{\left( ab \right)}^{n}}={{a}^{n}}{{b}^{n}}\]. We will use these properties to find the value of the square root of -16.

Complete step-by-step solution:
We are asked to find the square root of -16. We need to simplify and find its value. The given expression is of the form \[\sqrt{a}\], we know it can also be written as \[{{a}^{\dfrac{1}{2}}}\], here we have the value of a as -16. By doing this, we get \[{{\left( -16 \right)}^{\dfrac{1}{2}}}\].
Using the property \[{{\left( ab \right)}^{n}}={{a}^{n}}{{b}^{n}}\], we can express it as \[{{\left( -1 \right)}^{\dfrac{1}{2}}}{{\left( 16 \right)}^{\dfrac{1}{2}}}\]. We know that the square root of -1 is expressed as \[i\].
We know that 16 is a square of 4, thus we can write 16 as \[{{4}^{2}}\]. Putting this in the above expression we get \[i{{\left( {{4}^{2}} \right)}^{\dfrac{1}{2}}}\].
Using the algebraic property \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}\], we can express above expression as \[{{8}^{2\times \dfrac{1}{2}}}\]. Cancelling out the common factors, we get \[i4\].
Thus, the square root of -16 is \[i4\].

Note: One would normally think that we can not evaluate the square root of negative numbers. But here we used the imaginary numbers for it. Using the imaginary term \[i\] whose value equals square root of -1 algebraically expressed as \[{{\left( -1 \right)}^{\dfrac{1}{2}}}\], we can find square root of any negative number.