Let's simplify [tex]\(\sqrt{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x}\)[/tex] step-by-step.
1. To start, we'll express each radical in terms of exponents:
- [tex]\(\sqrt{x} = x^{\frac{1}{2}}\)[/tex]
- [tex]\(\sqrt[7]{x} = x^{\frac{1}{7}}\)[/tex].
Therefore, the given expression [tex]\(\sqrt{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x}\)[/tex] can be written as:
[tex]\[
x^{\frac{1}{2}} \cdot x^{\frac{1}{7}} \cdot x^{\frac{1}{7}}
\][/tex]
2. When multiplying expressions with the same base, we add their exponents:
[tex]\[
x^{\frac{1}{2}} \cdot x^{\frac{1}{7}} \cdot x^{\frac{1}{7}} = x^{\left(\frac{1}{2} + \frac{1}{7} + \frac{1}{7}\right)}
\][/tex]
3. Now, let's add the exponents. Sum up the fractions:
[tex]\[
\frac{1}{2} + \frac{1}{7} + \frac{1}{7}
\][/tex]
First, find a common denominator. The common denominator for 2 and 7 is 14:
[tex]\[
\frac{1}{2} = \frac{7}{14}
\][/tex]
[tex]\[
\frac{1}{7} = \frac{2}{14}
\][/tex]
Adding these fractions together:
[tex]\[
\frac{7}{14} + \frac{2}{14} + \frac{2}{14} = \frac{7 + 2 + 2}{14} = \frac{11}{14}
\][/tex]
4. Therefore, the exponent simplifies to [tex]\(\frac{11}{14}\)[/tex]:
[tex]\[
x^{\frac{1}{2} + \frac{1}{7} + \frac{1}{7}} = x^{\frac{11}{14}}
\][/tex]
Thus, the simplified form of [tex]\(\sqrt{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x}\)[/tex] is [tex]\(x^{\frac{11}{14}}\)[/tex].
Among the options given:
- [tex]\(x^{\frac{3}{7}}\)[/tex]
- [tex]\(x^{\frac{1}{7}}\)[/tex]
- [tex]\(x^{\frac{3}{21}}\)[/tex]
- [tex]\(\sqrt[21]{x}\)[/tex]
None of the provided options match our result. Therefore, based on the available choices, none of these is the correct simplified form for [tex]\(\sqrt{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x}\)[/tex]. The correct simplified form not listed among the options would be [tex]\(x^{\frac{11}{14}}\)[/tex].
