Solveeit Logo
Login

Question

Question: Simplify \({\left( { - 16} \right)^{\dfrac{{ - 1}}{2}}}\)?...

Simplify (−16)−12{\left( { - 16} \right)^{\dfrac{{ - 1}}{2}}}?

Explanation

Solution

In the given problem, we need to evaluate the inverse of square root of a given number. The given question requires knowledge of the concepts of complex numbers and square roots. The square root of a negative number is always a complex number. Hence, we must have in mind the definition of complex numbers and their basic properties.

Complete step by step answer:
In the question, we need to evaluate the value of (−16)−12{\left( { - 16} \right)^{\dfrac{{ - 1}}{2}}} . For evaluating the inverse of the square root of 1616, we need to have an idea of complex numbers, their properties and how to do basic operations on the complex numbers set.

So, (−16)−12{\left( { - 16} \right)^{\dfrac{{ - 1}}{2}}}

== (−116)12{\left( { - \dfrac{1}{{16}}} \right)^{\dfrac{1}{2}}}

(−116)12{\left( { - \dfrac{1}{{16}}} \right)^{\dfrac{1}{2}}} can be represented as a product of two entities: (−1)\left( { - 1} \right) and (116)\left( {\dfrac{1}{{16}}} \right). While splitting (−116)12{\left( { - \dfrac{1}{{16}}} \right)^{\dfrac{1}{2}}} into two entities, we need to distribute the power to both of them.

== (−1)12(116)12{\left( { - 1} \right)^{\dfrac{1}{2}}}{\left( {\dfrac{1}{{16}}} \right)^{\dfrac{1}{2}}}
Now, we know that the square root of 1616 is 44 since square root of any number is always positive.

== (−1)12(14){\left( { - 1} \right)^{\dfrac{1}{2}}}\left( {\dfrac{1}{4}} \right)

Now, we know that (−1)12=−1{\left( { - 1} \right)^{\dfrac{1}{2}}} = \sqrt { - 1} is known as iota and is represented as ii. So, the given expression can be represented as:

== i(14)=i4i\left( {\dfrac{1}{4}} \right) = \dfrac{i}{4}

So, (−16)−12{\left( { - 16} \right)^{\dfrac{{ - 1}}{2}}} can be simplified as (i4)\left( {\dfrac{i}{4}} \right) .

Note: The given question involves solving the square root of a negative number and that’s where the set of complex numbers comes into picture and plays a crucial role in mathematics.
The answer can be verified by working the solution backwards and observing that square of (i4)\left( {\dfrac{i}{4}} \right) is (−16)\left( { - 16} \right)