Question

Find the cube root of the following number by prime factorization method 175616.

Answer 1

Hint: For solving this problem, fast expand the given number into prime factors. Now, make groups of exponent 3 for the obtained prime factors. On taking the cube root, respective element is taken out from the pairs and then multiplied individually to obtain the final number.

Complete step-by-step solution -
Factors: All the numbers that divide a number completely, i.e., without leaving any remainder, are called factors of that number. One of the key expansions involved in mathematics is prime factorisation of a number. It includes representation of a number in terms of products of prime factors.
Now, proceeding to our problem, we are given a number 175616 and the cube root is required. Now, by using the prime factorisation, 175616 can be represented as:
\[\begin{align}
  & 175616=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 7\times 7\times 7 \\
 & 175616={{2}^{9}}\times {{7}^{3}} \\
 & 175616={{2}^{3}}\times {{2}^{3}}\times {{2}^{3}}\times {{7}^{3}} \\
\end{align}\]
Now, taking the cube root of both sides, we get
\[\begin{align}
  & \Rightarrow \sqrt[3]{175616}=\sqrt[3]{{{2}^{3}}\times {{2}^{3}}\times {{2}^{3}}\times {{7}^{3}}} \\
 & \Rightarrow {{\left( {{2}^{3}}\times {{2}^{3}}\times {{2}^{3}}\times {{7}^{3}} \right)}^{\dfrac{1}{3}}} \\
 & \Rightarrow {{2}^{\dfrac{3}{3}}}\times {{2}^{\dfrac{3}{3}}}\times {{2}^{\dfrac{3}{3}}}\times {{7}^{\dfrac{3}{3}}} \\
 & \Rightarrow 2\times 2\times 2\times 7 \\
 & \Rightarrow 56 \\
\end{align}\]
Therefore, the cube root of 175616 is 56 by using the prime factorisation technique.

Note: Another way for finding the cube root quickly is by hit and trial method. Unit digit of 175616 is 6. So, we can say that the unit digit of its cube root will be 6. Now, we find the cube root of 175616 by deriving from remaining digits. Let us consider the remaining digits leaving the last 3 digits. i.e. 175. Since 175 comes in between cubes of 5 and 6. So, the ten’s digit of the cube root will definitely be 5 i.e. cube root of 175616 will be 56.