Question

What is the square root of \[{{x}^{3}}\]?

Answer 1

Hint: First of all, let’s have a look at the basic definition of square root and cube of a function. The square root of any number is equal to a number, which when squared gives the original number, where we mean by square that the number obtained by multiplying the number with itself.

Complete step by step solution:
Here, \[\sqrt{x}\] is the radical symbol used to represent the square root of numbers. We can also write it as \[{{x}^{\dfrac{1}{2}}}\].
For example, the square of 3 is \[9,{{3}^{2}}=9\] and the square root of 9 is \[3,\sqrt{9}=3\] Since 9 is a perfect square, hence it is easy to find the square root of such numbers, but for an imperfect square like 3, 7, 5, etc., it is really tricky to find the root. In mathematics, a square root function is defined as a one-to-one function that takes a positive number as an input and returns the square root of the given input number.
If a number is a perfect square number, then there exists a perfect square root. If a number ends with 2, 3, 7 or 8 (in the unit digit), then the perfect square root does not exist.
And by cube root we simply mean the number multiplied by itself 3 times.
Now coming back to our question, we can see that we have to find the square root of cube of \[x\]So just simply writing cube of \[x\] in square root we get \[\sqrt{{{x}^{3}}}\]
We can simplify we get \[{{x}^{\dfrac{3}{2}}}\].

Note: Always remember that the square root of any negative numbers is not defined because the perfect square cannot be negative. When we multiply two negative numbers, then we get a positive number. We should recollect that square roots of negative numbers give us a complex quantity.