To rewrite the expression with rational exponents as a radical expression, we start with the given expression:
[tex]\[ 9 x^{\frac{3}{2}} \][/tex]
The goal is to convert this expression into its radical form. We can use the property of exponents, which states that:
[tex]\[ a^{\frac{m}{n}} = \sqrt[n]{a^m} \][/tex]
Let's apply this property to our expression.
1. Separate the coefficient from the variable with the exponent:
[tex]\[ 9 \cdot x^{\frac{3}{2}} \][/tex]
2. Convert the rational exponent [tex]\(\frac{3}{2}\)[/tex] into a radical form:
[tex]\[ x^{\frac{3}{2}} = \sqrt[2]{x^3} \][/tex]
Since the 2 in the denominator represents a square root, we can simplify the expression further:
[tex]\[ x^{\frac{3}{2}} = \sqrt{x^3} \][/tex]
3. Combine the coefficient 9 with the radical expression:
[tex]\[ 9 \cdot \sqrt{x^3} \][/tex]
So, the radical form of the expression [tex]\( 9 x^{\frac{3}{2}} \)[/tex] is:
[tex]\[ 9 \sqrt{x^3} \][/tex]
Therefore, the correct answer is:
[tex]\[ 9 \sqrt{x^3} \][/tex]
