Question

Find the cube root of \[729\times 216\]

Answer 1

Hint: We solve this problem by using the prime factorization method.
The prime factorization method is the method of representing the given number in the product of prime numbers.
We use prime factorization for both 729 and 216 so that we can represent the product as the cube of some other product.

Complete step by step answer:
We are asked to find the cube root of \[729\times 216\]
Let us assume that the given number as
\[\Rightarrow x=729\times 216\]
Now, let us take the numbers separately.
Let us take the umber 729 and use the prime factorisation with 3 then we get
\[\Rightarrow 729=3\times 243\]
Again by using the prime number 3 to 243 then we get
\[\Rightarrow 729={{3}^{2}}\times 81\]
We know that 81 can be represented as \[{{3}^{4}}\]
By using this representation in above equation then we get
\[\begin{align}
  & \Rightarrow 729={{3}^{2}}\times {{3}^{4}} \\
 & \Rightarrow 729={{3}^{6}} \\
\end{align}\]
Now, let us take the second number 216
Now, by using the prime factorisation by using the first prime number 2 then we get
\[\Rightarrow 216=2\times 108\]

Again by using the prime number 2 to 108 then we get
\[\Rightarrow 216={{2}^{2}}\times 54\]
Again by using the prime number 2 to 54 then we get
\[\Rightarrow 216={{2}^{3}}\times 27\]
We know that 27 can be represented as \[{{3}^{3}}\]
By using this representation in above equation then we get
\[\Rightarrow 216={{2}^{3}}\times {{3}^{3}}\]
Now, let us use the prime factorisation of individual number in the original product then we get
\[\begin{align}
  & \Rightarrow x=729\times 216 \\
 & \Rightarrow x={{3}^{6}}\times {{2}^{3}}\times {{3}^{3}} \\
 & \Rightarrow x={{2}^{3}}\times {{3}^{9}} \\
\end{align}\]
Now, let us represent the above product as the cube of some other product then we get
\[\Rightarrow x={{\left( 2\times {{3}^{3}} \right)}^{3}}\]
Now, by applying the cube root on both sides then we get
\[\begin{align}
  & \Rightarrow \sqrt[3]{x}=2\times {{3}^{3}} \\
 & \Rightarrow \sqrt[3]{x}=2\times 27=54 \\
\end{align}\]
Therefore, we can conclude that the cube root of \[729\times 216\] is 54

Note:
We can solve this problem by using the exponent formula.
We have the formula of cube roots of the product of two number as
\[\Rightarrow \sqrt[3]{ab}=\sqrt[3]{a}\times \sqrt[3]{b}\]
By using the above formula for a given number then we get
\[\Rightarrow \sqrt[3]{729\times 216}=\sqrt[3]{729}\times \sqrt[3]{216}\]
We know that the cube root of 729 is 9 and the cube root of 216 is 6
By using the above condition we get
\[\Rightarrow \sqrt[3]{729\times 216}=9\times 6=54\]
Therefore, we can conclude that the cube root of \[729\times 216\] is 54