Question

How do you write $\sqrt {{x^5}}$ in exponential form?

Answer 1

In this question, we need to write $\sqrt {{x^5}}$ in exponential form. Then we will use a law of radicals, to write the given term into the simplest form of exponent. The law states that, if $n$ is a positive integer that is greater than n and a is a real number or a factor, then ${\left( a\right)^{\dfrac{x}{n}}} = \sqrt[n]{{{a^x}}}$ . Then we will substitute the values and determine the exponential form of $\sqrt {{x^5}}$.

Complete step-by-step solution:
Here, we need to write to convert $\sqrt {{x^5}}$ in exponential form.
Expressing in simplest radical form is nothing but simplifying the radical into the simplest form with no more square roots, cube roots, etc. left to find. In other words, a number under a radical is indivisible by a perfect square other than $1$.
If n is a positive integer that is greater than x and a is a real number or a factor, then we get,
${\left( a \right)^{\dfrac{x}{n}}} = \sqrt[n]{{{a^x}}}$
It means raise a to the power x then find the nth root of the result.
So, we can write $\sqrt[n]{{{a^x}}}$ as ${\left( a \right)^{\dfrac{x}{n}}}$.
Hence, the given expression in radical form $\sqrt {{x^5}}$ can also be written as ${\left( x \right)^{\dfrac{5}{2}}}$ in exponential form.
Hence, the required answer is ${\left( x \right)^{\dfrac{5}{2}}}$ .

Note: In this question it is important to note here that we used a law of the radical to solve this form i.e., a radical represents a fractional exponent in which the numerator of the fractional exponent is the power of the base and the denominator of the fractional exponent is the index of the radical.