Question

Find the cube root of 343.

Answer 1

Hint: Here, to find the cube root of 343 we can use the method of prime factorisation. In prime factorisation we factorise the numbers into prime numbers, called as prime factors. After doing a prime factorisation group the factors in three such that each member of the group is the same. Take one factor from each group and then multiply to obtain the cube root.

Complete step-by-step answer:

Here, we have to find the cube root of 343.

First let us discuss the cube root. The cube root of a number is a special value that, when used in multiplication three times gives that number. That is, if ‘$a$’ is the cube root of $x$ then we can write $x={{a}^{3}}=a\times a\times a$.

Here, to find the cube root of 343 we can use the method of prime factorisation.

In prime factorisation we factorise the numbers into prime numbers, called as prime factors. There are two methods of prime factorisation.

Division method
Factor tree method

 Here, we are applying the division method. In the division method of prime factorisation, the following steps should be followed.

First we divide the number by the smallest prime number which divides the number exactly.
We divide the quotient again by the smallest prime number, or the next smallest prime number, if it is not exactly divisible by the smallest prime number. Repeat the process again and again till the quotient becomes 1. Here we are using only prime numbers to divide.

We multiply all the prime factors, where the product is the number itself.

Therefore, for finding cube root by prime factorisation we have to group the factors in three such that each number of the group is the same. Take one factor from each group and then multiply to obtain the cube root.

 Now let us consider 343. First we have to do the prime factorisation:

$343=7\times 7\times 7$

So here, only one group is there with all three members as 7. Therefore 7 is the cube root.

$\begin{align}
  & 3\sqrt[{}]{343}=3\sqrt[{}]{7\times 7\times 7} \\
 & 3\sqrt[{}]{343}=7 \\
\end{align}$

Hence, we can say that the cube root of 343 is 7.

Note: Here, after the prime factorisation you should not stop. Group the factors in three of the same numbers and then multiply one factor from each group. That is, here the factors are only 7 and 7 is repeated 3 times. Therefore, you have to take the cube root as 7. You can recheck the answer by multiplying 7 three times.