Question

Find the cube of $ {{\left( 21 \right)}^{\dfrac{2}{7}}} $ ?

Answer 1

Hint: We know that taking a cube of a number is equal to taking 3 in the power of that number. The number given in the above problem is equal to $ {{\left( 21 \right)}^{\dfrac{2}{7}}} $ so taking a cube of this number means taking 3 in the power of that number. After putting 3 as power of $ {{\left( 21 \right)}^{\dfrac{2}{7}}} $ we are going to split 21 as $ 3\times 7 $ and then simplify.

Complete step by step answer:
The number given above of which we have to take the cube is as follows:
 $ {{\left( 21 \right)}^{\dfrac{2}{7}}} $
Now, we are going to take the cube of the above number by putting 3 in the power of the above number.
 $ {{\left( {{\left( 21 \right)}^{\dfrac{2}{7}}} \right)}^{3}} $
We can simplify the above expression by multiplying 3 by $ \dfrac{2}{7} $ and we get,
\[\begin{align}
  & {{\left( 21 \right)}^{\dfrac{2}{7}\times 3}} \\
 & ={{\left( 21 \right)}^{\dfrac{6}{7}}} \\
\end{align}\]
We can also simplify the above expression by splitting 21 as $ 3\times 7 $ .
 $ {{\left( 7\times 3 \right)}^{\dfrac{6}{7}}} $
There is a property of exponents that:
 $ {{\left( a\times b \right)}^{c}}={{a}^{c}}\times {{b}^{c}} $
Using the above property of exponents in simplifying $ {{\left( 7\times 3 \right)}^{\dfrac{6}{7}}} $ we get,
 $ {{7}^{\dfrac{6}{7}}}\times {{3}^{\dfrac{6}{7}}} $
From the above, we got the cube of $ {{\left( 21 \right)}^{\dfrac{2}{7}}} $ as:
 $ {{7}^{\dfrac{6}{7}}}\times {{3}^{\dfrac{6}{7}}} $

Note:
 In the above solution, we have just used the property that while taking a cube of any number we will take 3 in the power of the number. In the below, we are showing how cubing a number is equal to taking 3 as the power of the given number.
Cube of any number is the multiplication of the same number by 3 times. Let us take a number say “a” so cube of this number is equal to the multiplication of “a” by 3 times which is shown below.
 $ a\times a\times a $
Now, there is a property of the exponents that:
 $ {{p}^{l}}\times {{p}^{m}}\times {{p}^{n}} $
When base is same i.e. “p” then we can add the powers of the same base and the above expression will look like:
 $ {{p}^{l+m+n}} $
Now, we are going to use this property in simplifying the above expression i.e. $ a\times a\times a $ . As the power of “a” is 1 so adding all the three powers of “a” will give
 $ {{a}^{3}} $