Question

rational exponent and radical forms
consider the expression ( x^{3/2} ).
what is the radical form of the given expression?
a ( sqrt4{x^{3/2}} )
b ( sqrt3{x^2} )
c ( sqrt{x^{2/3}} )
d ( sqrt{x^3} )
what is the rational exponential form of ( sqrt3{x^2} )

Explanation:

Step1: Recall the formula for converting rational exponents to radicals

The formula is \( a^{\frac{m}{n}}=\sqrt[n]{a^{m}} \) (where \( n\gt0 \), \( n\in\mathbb{N} \), \( m\in\mathbb{Z} \)). For the expression \( x^{\frac{3}{2}} \), here \( a = x \), \( m = 3 \), \( n = 2 \).

Step2: Apply the formula

Using the formula \( a^{\frac{m}{n}}=\sqrt[n]{a^{m}} \), substitute \( a=x \), \( m = 3 \), \( n = 2 \). We get \( x^{\frac{3}{2}}=\sqrt[2]{x^{3}}=\sqrt{x^{3}} \).

Answer:

D. \(\sqrt{x^{3}}\)