Question

How do you write the expression $ {6^{\dfrac{3}{2}}} $ in radical form ?

Answer 1

Hint: In order to write the expression into the radical form , factorize the base part of the exponent such that it contains perfect squares in it and then use laws of exponents to get the required answer.
Formula:
 $ {(a)^{\dfrac{m}{n}}} $ = $ {({a^m})^{\dfrac{1}{n}}} $
 $ {x^{m + n}} = {x^m} \times {x^n} $

Complete step-by-step answer:
Given a number with the exponent value
 $ = {6^{\dfrac{3}{2}}} $
Separating the exponent value by using the identity of exponents that $ {(a)^{\dfrac{m}{n}}} $ can be written as $ {({a^m})^{\dfrac{1}{n}}} $
Now rewriting the question
 $
   = {({6^3})^{\dfrac{1}{2}}} \\
   = {(216)^{\dfrac{1}{2}}} \;
  $
Now to convert the above number into radical form factorize the value in the bracket such that it has some perfect squares
 $
   = {(6 \times 6 \times 6)^{\dfrac{1}{2}}} \\
   = 6{(6)^{\dfrac{1}{2}}} \;
   = 6\sqrt 6 \;
  $
Therefore , $ {6^{\dfrac{3}{2}}} $ in radical form is $ 6\sqrt 6 $
Alternative:
Alternatively to determine the radical form of the above question we can do the question by the exponent value as
 $
   = {6^{\dfrac{3}{2}}} \\
   = {6^{\dfrac{1}{2} + 1}} \\
  $
And using exponent identity $ {x^{m + n}} = {x^m} \times {x^n} $
 $
   = {6^1} \times {6^{\dfrac{1}{2}}} \\
   = 6\sqrt 6 \;
  $
So, the correct answer is “ $ 6\sqrt 6 $ ”.

Note: A radical represents a fractional exponent in which the numerator of the fractional exponent is the power of the base and the denominator of the fractional exponent is the index of the radical. Product: The nth root of a product is equal to the product of the nth root of each factor