Question

How do you simplify ${{x}^{\dfrac{2}{5}}}$?

Answer 1

Hint: In this question we have an exponential expression which has a power which is in terms of a fraction therefore, we will use the required exponential properties and convert the expression into the form of a radical expression which consists of a root term.

Complete step-by-step answer:
We have the term given to us as:
$\Rightarrow {{x}^{\dfrac{2}{5}}}$
We can see that we have a single term which in the form of an exponent which is in the form of a fraction therefore we can use the property of exponents which is ${{a}^{\dfrac{b}{c}}}={{({{a}^{b}})}^{\dfrac{1}{c}}}=\sqrt[c]{{{p}^{b}}}$.
On using the property, we get:
$\Rightarrow {{\left( {{x}^{2}} \right)}^{\dfrac{1}{5}}}$
Now since in the denominator the term is $5$, we will take the fifth-root of the term. The fifth-root implies the number which has to be multiplied by itself to get the original number. Writing it in the form of the radical expression, we get:
$\Rightarrow \sqrt[5]{{{x}^{2}}}$, which is the required solution.

Note: It is to be noted that the function can be read out as ‘ $5th$ root of $x$ raised to $2$’. This means that whatever is the solution when the term $x$ is raised to the power of $2$, we have to take the fifth root of the number, which means the number which has to be multiplied five times to get ${{x}^{2}}$.
Exponents are used to write the similar terms in multiplication in a simple format. Some exponential numbers are too big to be calculated directly, therefore to solve them logarithm is used. Logarithm is used in exponents to convert the exponential term in the form of multiplication. The antilog of the term is then taken which is the inverse of taking log, to get the required solution back after simplification.