Question

rational exponent and radical forms
consider the expression ( x^{3/2} )

what is the radical form of the given expression?
a ( sqrt3{x^{2/3}} )
b ( sqrt3{x^2} )
c ( sqrt{x^{2/3}} )
d ( sqrt{x^3} )

what is the rational exponential form of ( sqrt3{x^2} )?
a ( 2x^3 )
b ( 3x^2 )
c ( x^{2/3} )
d ( x^{3/2} )

Explanation:

First Question (Radical Form of \( x^{3/2} \))

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

For the expression \( x^{3/2} \), here \( a = x \), \( m = 3 \), and \( n = 2 \).

Step2: Apply the formula. Using \( a^{m/n} = \sqrt[n]{a^m} \), we substitute the values: \( x^{3/2} = \sqrt[2]{x^3} \) (since the square root has an index of 2, we can write it as \( \sqrt{x^3} \)). Now let's check the options:

• Option A: \( \sqrt[3]{x^{2/3}} \) does not match.
• Option B: \( \sqrt[3]{x^2} \) has index 3 and exponent 2, not matching.
• Option C: \( \sqrt{x^{2/3}} \) has exponent \( 2/3 \) inside, not matching.
• Option D: \( \sqrt{x^3} \) matches our result.

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

For the expression \( \sqrt[3]{x^2} \), here \( a = x \), \( m = 2 \), and \( n = 3 \).

Step2: Apply the formula. Using \( \sqrt[n]{a^m} = a^{m/n} \), we substitute the values: \( \sqrt[3]{x^2} = x^{2/3} \)? Wait, no, wait the options: Wait, maybe I misread. Wait the radical is \( \sqrt[3]{x^2} \)? Wait no, looking at the second question: "What is the rational exponential form of \( \sqrt[3]{x^2} \)?" Wait the options: A: \( 2x^3 \), B: \( 3x^2 \), C: \( x^{3/3} \) (wait no, C is \( x^{2/3} \)? Wait the user's image: Wait maybe a typo, but let's re-express. Wait the radical is \( \sqrt[3]{x^2} \), so using \( \sqrt[n]{a^m}=a^{m/n} \), so \( \sqrt[3]{x^2}=x^{2/3} \)? But the options: Wait the options given are A: \( 2x^3 \), B: \( 3x^2 \), C: \( x^{2/3} \) (maybe the user wrote \( x^{3/3} \) by mistake, but likely it's \( x^{2/3} \))? Wait no, looking at the options again:

Wait the second question's options:

A: \( 2x^3 \)

B: \( 3x^2 \)

C: \( x^{2/3} \) (maybe the user's image has a typo, but let's check the formula again. Wait, if the radical is \( \sqrt[3]{x^2} \), then \( n = 3 \), \( m = 2 \), so \( x^{2/3} \), which is option C (if C is \( x^{2/3} \), maybe the user wrote \( x^{3/3} \) by mistake, but likely it's \( x^{2/3} \)). Wait, maybe I misread the radical. Wait the second question: "What is the rational exponential form of \( \sqrt[3]{x^2} \)?" So applying \( \sqrt[n]{a^m}=a^{m/n} \), we get \( x^{2/3} \), which is option C (assuming C is \( x^{2/3} \), maybe the user's image has a typo in the exponent, but based on the formula, that's the correct form.

Answer:

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

Second Question (Rational Exponent Form of \( \sqrt[3]{x^2} \))