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If the cube of a is equal to b, then cube root of b=
A. ${{a}^{9}}$
B. ${{a}^{3}}$
C. $a$
D. ${{b}^{3}}$

Last updated date: 19th Jun 2024
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Hint: We have to find the cube root of ‘b’ in terms of ‘a’. We have been given that ‘b’ is the cube of ‘a’. Therefore, we will form an equation in ‘a’ and ‘b’ using this relation and write ‘b’ in terms of ‘a’. Once we are done doing that, we will raise the power to $\dfrac{1}{3}$ on both sides, i.e., we will take cube root on both sides of the equation. Taking the cube root on both sides, we will find the cube root of ‘b’ in terms of ‘a’. Hence we will have to find out the answer.

Complete step-by-step solution:
Now, we have been given that cube of ‘a’ is equal to ‘b’.
Therefore, we can write this as:
Thus, ‘b’ in terms of ‘a’ can be written as:
$b={{a}^{3}}$ .........................(1)
Now, that we have written ‘b’ in terms of ‘a’, we can take cube root on both sides to find our answer.
After taking cube root on both sides of equation (1) we will get:
  & b={{a}^{3}} \\
 & \Rightarrow {{b}^{\dfrac{1}{3}}}={{\left( {{a}^{3}} \right)}^{\dfrac{1}{3}}} \\
Now, we will use the property ${{\left( {{x}^{m}} \right)}^{n}}={{x}^{mn}}$
Using this property we will get:
  & \Rightarrow {{b}^{\dfrac{1}{3}}}={{\left( a \right)}^{3\times \dfrac{1}{3}}} \\
 & \Rightarrow {{b}^{\dfrac{1}{3}}}={{a}^{1}} \\
 & \Rightarrow {{b}^{\dfrac{1}{3}}}=a \\
Hence, the cube root of ‘b’ is ‘a’.
Thus, option (C) is the correct option.

Note: Please keep in mind the signs of the numbers, i.e. negative or positive numbers, while taking cube roots on both sides because a negative number always has a negative cube root and a negative cube. Here, the signs are both taken as positive hence this aspect is not considered here.