Question

How do you find the cube root of $250$

Answer 1

Hint: In this problem we need to calculate the cube root of the given number. It will be very easy to calculate cube roots when we have the given number in exponential form. To write the given number in exponential form we are going to use the prime factorization form of the given number. From the prime factorisation form of the given number, we will convert it to exponential form by using the exponential formula $a\times a\times a\times a.....\text{n times}={{a}^{n}}$. After getting exponential form to get the cube root of the number, calculate the ${{\left( \dfrac{1}{3} \right)}^{rd}}$ power of the given number. Here we will use the exponential formula ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}$ and simplify the equation to get the required result.

Complete step by step solution:
Given number is $250$.
We can write the number $250$ in prime factorization form as
$250=2\times 5\times 5\times 5$
Applying the exponential form $a\times a\times a\times a.....\text{n times}={{a}^{n}}$ in the above equation, then we will get
$250=2\times {{5}^{3}}$
To find the cube root of the given number we need to calculate the ${{\left( \dfrac{1}{3} \right)}^{rd}}$ power of the number. We can write this mathematically as
$\sqrt[3]{250}={{\left( 2\times {{5}^{3}} \right)}^{\dfrac{1}{3}}}$
Applying the exponential formula ${{\left( ab \right)}^{m}}={{a}^{m}}\times {{b}^{m}}$ in the above equation, then we will have
$\sqrt[3]{250}={{2}^{\dfrac{1}{3}}}\times {{\left( {{5}^{3}} \right)}^{\dfrac{1}{3}}}$
Applying the exponential formula ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}$ in the above equation, then we will get
$\sqrt[3]{250}={{2}^{\dfrac{1}{3}}}\times {{\left( 5 \right)}^{3\times \dfrac{1}{3}}}$
Simplifying the above value, then we will have
$\sqrt[3]{250}=5\sqrt[3]{2}$

Hence the cube root of the number $250$ is $5\sqrt[3]{2}$.

Note: In the problem they haven't asked to calculate the exact value for the cube root of the given number. So, we have calculated the simple value of the cube root. If they have asked to calculate the exact value then we will use the value $\sqrt[3]{2}=1.2599$ in the value $5\sqrt[3]{2}$, then we will get
$\begin{align}
  & \sqrt[3]{250}=5\left( 1.2599 \right) \\
 & \Rightarrow \sqrt[3]{250}=6.2995 \\
\end{align}$
Hence the exact value of cube root of $250$ is $6.2995$.