Question

Find the cube root of 729 $ \times $ 216.

Answer 1

Hint: In this question , first we have to know what cube root is . A cube root is a number that can be multiplied by itself three times to equal the original value. Finding a cube root is the opposite of cubing a number or raising a number to the third power. For ex : cube root of $y$ will be $x$ if $y = x \times x \times x$.

Complete step-by-step solution -
Here ,we have to find out the cube root of 729 $ \times $ 216
We will find the cube root of 729 and 216 separately and then we will multiply them to get the required answer.
So in our case we need to figure out what two numbers will multiply by itself three times to give us 729 and \[216\].

1. Let us first find the cube root of 729 .
$
   \Rightarrow 729 = 3 \times (243) \\
   \Rightarrow 729 = 3 \times 3 \times (81) \\
   \Rightarrow 729 = 3 \times 3 \times 3 \times (27) \\
   \Rightarrow 729 = 3 \times 3 \times 3 \times 3 \times (9) \\
   \Rightarrow 729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \\
$
We cannot factorise this more , it can also be written as :
 $ \Rightarrow 729 = (3 \times 3) \times (3 \times 3) \times (3 \times 3).$
$ \Rightarrow 729 = 9 \times 9 \times 9$
Hence, we can clearly see that when we multiply 9 by itself three times we will get 729.
Therefore, the cube root of 729, $\sqrt[3]{{729}} = 9$.

2. Now we will find the cube root of 216.
$
   \Rightarrow 216 = 2 \times (108) \\
   \Rightarrow 216 = 2 \times 2 \times (54) \\
   \Rightarrow 216 = 2 \times 2 \times 2 \times (27) \\
$
$
   \Rightarrow 216 = 2 \times 2 \times 2 \times 3 \times (9) \\
   \Rightarrow 216 = 2 \times 2 \times 2 \times 3 \times 3 \times 3 \\
$
We cannot factorise this more , hence
We can also write this as :
$
   \Rightarrow 216 = (2 \times 3) \times (2 \times 3) \times (2 \times 3) \\
   \Rightarrow 216 = 6 \times 6 \times 6 \\
$
Here we can clearly see that , when we multiply 6 by itself three times ,we will get \[216\]
Therefore , cube root of \[216\], $\sqrt[3]{{216}} = 6$.
So now we will multiply the cube roots of both 729 and 216 ,
$
   \Rightarrow \sqrt[3]{{729 \times \;216{\text{ }}}} = \sqrt[3]{{{{(9)}^3} \times {{(6)}^3}}} \\
   \Rightarrow \sqrt[3]{{729 \times \;216{\text{ }}}} = 9 \times 6 \\
   \Rightarrow \sqrt[3]{{729 \times \;216{\text{ }}}} = 54 \\
$
Hence, the cube root of 729 $ \times $ 216 is 54.

Note: In this type of question, we will simply use the method of factorization. We have given a number and we will make factors of that number in such a way that when we multiply one of its common factors by itself three times, we will get the required number. Then we will find out the cube roots of both numbers separately and then by simply multiplying them according to the question we will get our answer.