Question

How do you write $${\left(125\right)}^{{\displaystyle \frac{1}{3}}}$$?

Answer 1

Hint: To solve the given question we will need the following properties. The property of exponents which states that $${\left(a^m\right)}^n=a^{mn}$$ here, $$a,m,n\in$$Real numbers. We should also know that if a is the cube of b, then it can be written as $$a=b^3$$, we will use these properties to solve the given question.

Complete step by step solution:
We are asked to simplify $${\left(125\right)}^{{\displaystyle \frac{1}{3}}}$$, which means we have to find its value. We know that 125 is the cube of 5. Using the property which states that, if a is the cube of b, then it can be written as $$a=b^3$$. Here we have a = 125, and b = 5. By substituting the value, it can be written as, $$125=5^3$$. Using this in the given question, it can be simplified as,
$$\begin{array}{rl}& {\left(125\right)}^{{\displaystyle \frac{1}{3}}}\\ & \Rightarrow {\left(5^3\right)}^{{\displaystyle \frac{1}{3}}}\end{array}$$
We know the property of exponents which states that, $${\left(a^m\right)}^n=a^{mn}$$ here, $$a,m,n\in$$ Real numbers. We have a = 5, m = 3, and $$n={\displaystyle \frac{1}{3}}$$. Using this property in the above expression it becomes,
$$\begin{array}{rl}& \Rightarrow {\left(5^3\right)}^{{\displaystyle \frac{1}{3}}}\\ & \Rightarrow 5^{3\times {\displaystyle \frac{1}{3}}}=5\end{array}$$

Hence the given expression $${\left(125\right)}^{{\displaystyle \frac{1}{3}}}$$ can be written as 5.

Note: These types of questions can be solved by remembering the values of squares, cubes, square roots, and cube roots of the numbers. We can also solve this question by factorization of the number 125, and then taking the factors out of the cube root which appears thrice. This can be done as follows, we know that $${\left(125\right)}^{{\displaystyle \frac{1}{3}}}$$ can also be written as $$\sqrt[3]{125}$$. Hence, $$\sqrt[3]{125}=\sqrt[3]{5\times 5\times 5}=5$$. We can see that we are getting the same answer from both methods.