Certainly! Let's rewrite the expression [tex]\( 7x^{\frac{2}{3}} \)[/tex] as a radical expression step by step.
1. Identify the given expression:
The given expression is [tex]\( 7x^{\frac{2}{3}} \)[/tex].
2. Recall the property of exponents with rational numbers:
The expression [tex]\( x^{\frac{a}{b}} \)[/tex] can be rewritten in radical form as [tex]\( \sqrt[b]{x^a} \)[/tex].
3. Apply the property to the given expression:
Here [tex]\( a = 2 \)[/tex] and [tex]\( b = 3 \)[/tex]. So, [tex]\( x^{\frac{2}{3}} \)[/tex] can be rewritten as [tex]\( \sqrt[3]{x^2} \)[/tex].
4. Include the constant multiplicative factor:
The given expression is [tex]\( 7x^{\frac{2}{3}} \)[/tex]. Using the result from step 3, we rewrite it as:
[tex]\[
7x^{\frac{2}{3}} = 7 \cdot \sqrt[3]{x^2}
\][/tex]
Hence, the expression [tex]\( 7x^{\frac{2}{3}} \)[/tex] rewritten with rational exponents as a radical expression is [tex]\( 7 \sqrt[3]{x^2} \)[/tex].
Among the given options:
1. [tex]\( \sqrt[3]{7x^2} \)[/tex]
2. [tex]\( \sqrt{7x^3} \)[/tex]
3. [tex]\( 7 \sqrt[3]{x^2} \)[/tex]
4. [tex]\( 7 \sqrt{x^3} \)[/tex]
The correct option is:
[tex]\[ \boxed{3. \ 7 \sqrt[3]{x^2}} \][/tex]
