To represent [tex]\( 6x^{\frac{3}{4}} \)[/tex] in radical form, let's break it down step by step:
1. Understand the exponent [tex]\( \frac{3}{4} \)[/tex] associated with [tex]\( x \)[/tex]:
- The exponent [tex]\( \frac{3}{4} \)[/tex] means [tex]\( x \)[/tex] raised to the power of 3, and then taking the 4th root of the result. This can be written as:
[tex]\[
x^{\frac{3}{4}} = (x^3)^{\frac{1}{4}}
\][/tex]
2. Convert [tex]\( x^{\frac{3}{4}} \)[/tex] to its radical form:
- The notation [tex]\( (x^3)^{\frac{1}{4}} \)[/tex] is the same as taking the 4th root of [tex]\( x^3 \)[/tex]. In radical form, this is written as:
[tex]\[
\sqrt[4]{x^3}
\][/tex]
3. Incorporate the constant multiplying the expression:
- The original expression is [tex]\( 6x^{\frac{3}{4}} \)[/tex], so we need to include the constant 6 in our radical form:
[tex]\[
6 \cdot \sqrt[4]{x^3}
\][/tex]
4. Match the derived expression with the given options:
- Comparing [tex]\( 6 \cdot \sqrt[4]{x^3} \)[/tex] with the provided choices, we find that this matches with:
[tex]\[
6 \sqrt[4]{x^3}
\][/tex]
Therefore, the correct representation of [tex]\( 6x^{\frac{3}{4}} \)[/tex] in radical form is:
[tex]\[
6 \sqrt[4]{x^3}
\][/tex]
So, the answer is [tex]\(\boxed{1}\)[/tex].
