Question

How do you find the cube root of $729$ ?

Answer 1

Hint: For finding the cube root we use prime factorization method. In this method we factorize the numbers into prime factors and then group the factors in 3 and take one factor from each and find the product of the factors.

Complete step-by-step answer:
The objective of the problem is to find the value of $\sqrt[3]{{729}}$
To find the value of $\sqrt[3]{{729}}$ we use long division method of prime factorization
Factorization by long division method :
In this method firstly we write the given number into prime factors. In the next step we write obtained prime factors as a group of three as it is cube root the grouped numbers should be the same. Then take out the one factor from the group and find the product of obtained factors. The obtained product is our answer to the given cube root number.
Now let us find the cube root of 729 by following the above steps.
The least prime factor of 729 is three.
$\begin{gathered}
  3\left| \!{\underline {\,
  {729} \,}} \right. \\
  3\left| \!{\underline {\,
  {243} \,}} \right. \\
  3\left| \!{\underline {\,
  {81} \,}} \right. \\
  3\left| \!{\underline {\,
  {27} \,}} \right. \\
  3\left| \!{\underline {\,
  9 \,}} \right. \\
  3\left| \!{\underline {\,
  3 \,}} \right. \\
  \,\,\,1 \\
\end{gathered} $
In the above step we divide 729 by three and the quotient that obtained after dividing is written below as 243 again we are dividing 243 by three and the quotient 81 is written below. We follow this until we get one.
Now we express the given number into a product of prime numbers.
$729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 1$
Now grouping the factors as three .
$729 = \left( {3 \times 3 \times 3} \right) \times \left( {3 \times 3 \times 3} \right) \times 1$
Now take the cube root on both sides we get $\sqrt[3]{{729}} = \sqrt[3]{{\left( {3 \times 3 \times 3} \right) \times \left( {3 \times 3 \times 3} \right) \times 1}}$
On taking out one factor from each group , we get
$\sqrt[3]{{729}} = 3 \times 3 \times 1$
On multiplying the factors we get ,
\[\sqrt[3]{{729}} = 9\]
Hence the square root of 729 is 9.
The given number is a perfect cube.

Note: The prime factors of a number are the prime numbers that when multiplied it gives the original number. A perfect cube is a number where the product of some integer with itself three times gives the cube number.