Question

rational exponent and radical forms
consider the expression ( x^{3/2} ).
options (partial):
b ( sqrt3{x^2} )
c ( sqrt{x^{2/3}} )
d ( sqrt{x^3} )
what is the rational exponential form of ( sqrt3{x^2} )?
options:
a ( 2x^3 )
b ( 3x^2 )
c ( x^{2/3} )
d ( x^{3/2} )

Explanation:

Step1: Recall the formula for converting radicals to rational exponents. The formula is $\sqrt[n]{a^m} = a^{\frac{m}{n}}$, where $n$ is the index of the radical and $m$ is the exponent of the base inside the radical.

Step2: Apply the formula to $\sqrt[3]{x^2}$. Here, $n = 3$ (the index of the cube root) and $m = 2$ (the exponent of $x$ inside the radical). So, using the formula $\sqrt[n]{a^m} = a^{\frac{m}{n}}$, we substitute $a = x$, $m = 2$, and $n = 3$. This gives us $x^{\frac{2}{3}}$.

Answer:

C. $x^{2/3}$