Question

Calculate the cube root of \[54\].

Answer 1

Hint: When a number is multiplied by itself three times the result thus obtained is called the cube of the number. Hence, when a number is raised to the power \[\dfrac{1}{3}\], the result obtained is the cube root of the number, which means the cube root of \[x\] is \[{x^{\dfrac{1}{3}}}\].

Complete step by step solution:
To find the cube root of a number first factorize it, to find the prime factors. Factorization is the process of writing a number as a product of several factors. Prime factorization means to factorize the number in such a way so as to obtain only the prime factors.
Divide \[54\] by the smallest prime number \[2\] the quotient is \[27\], again divide the quotient by another prime number. But \[27\] is not divisible by \[2\] so divide it by \[3\], now the quotient is \[9\], again divide \[9\] by \[3\], the quotient is \[3\] finally divide \[3\] by \[3\] to get the quotient \[1\].
Thus the prime factorization of \[54\] can be written as:
\[54 = 2 \times 3 \times 3 \times 3\]
Next, group the factors in groups of triplets. Observe that only \[3\] forms a triplet that is it appears thrice and \[2\] appears only once, hence the factors of \[54\] can be written as:
\[54 = {3^3} \times 2\]
\[ \Rightarrow \sqrt[3]{{54}} = \sqrt[3]{{\left( {{3^3} \times 2} \right)}}\]
\[ \Rightarrow \sqrt[3]{{54}} = \sqrt[3]{{{3^3}}} \times \sqrt[3]{2}\]
\[ \Rightarrow \sqrt[3]{{54}} = 3 \times \sqrt[3]{2}\]
\[ \Rightarrow \sqrt[3]{{54}} = 3\sqrt[3]{2}\]
Substitute the value of \[\sqrt[3]{2}\], i.e. \[1.259\]
\[ \Rightarrow \sqrt[3]{{54}} = 3 \times 1.259\]
\[ \Rightarrow \sqrt[3]{{54}} = 3.77\]

Hence the value of the cube root of \[54\] is \[3.77\].

Note:
Prime numbers are numbers that can be divided only by one and itself. Example: \[2,3\].
The cube root of a number can also be found by the method of estimation. But the factorization method is better than the estimation as it has fewer calculations and is faster too.