To simplify the expression [tex]\(7 \sqrt{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x}\)[/tex], we need to break down each component and then combine them appropriately.
First, recall the definition of each radical expression in terms of exponents:
- [tex]\(\sqrt{x}\)[/tex] can be written as [tex]\(x^{\frac{1}{2}}\)[/tex].
- [tex]\(\sqrt[7]{x}\)[/tex] can be written as [tex]\(x^{\frac{1}{7}}\)[/tex].
Now we rewrite the given expression using these exponent forms:
[tex]\[7 \sqrt{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x}\][/tex]
[tex]\[= 7 \cdot x^{\frac{1}{2}} \cdot x^{\frac{1}{7}} \cdot x^{\frac{1}{7}}\][/tex]
Next, we combine the exponents of [tex]\(x\)[/tex]. When multiplying expressions with the same base, we add their exponents:
[tex]\[ 7 \cdot x^{\left(\frac{1}{2} + \frac{1}{7} + \frac{1}{7}\right)} \][/tex]
Let's find the sum of the exponents step-by-step:
1. Simplify [tex]\(\frac{1}{7} + \frac{1}{7}\)[/tex]:
[tex]\[
\frac{1}{7} + \frac{1}{7} = \frac{2}{7}
\][/tex]
2. Add this result to [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[
\frac{1}{2} + \frac{2}{7}
\][/tex]
To sum these fractions, we need a common denominator. The least common multiple of 2 and 7 is 14. Convert each fraction:
[tex]\[
\frac{1}{2} = \frac{7}{14}
\][/tex]
[tex]\[
\frac{2}{7} = \frac{4}{14}
\][/tex]
Now add these converted fractions:
[tex]\[
\frac{7}{14} + \frac{4}{14} = \frac{11}{14}
\][/tex]
Thus, the combined exponent is [tex]\(\frac{11}{14}\)[/tex]. Therefore, we have:
[tex]\[
7 \cdot x^{\frac{11}{14}}
\][/tex]
Since the problem asks for the simplified form involving just the exponents of [tex]\(x\)[/tex], the correct answer is:
[tex]\[ x^{\frac{11}{14}} \][/tex]
Now comparing with the given options:
- [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 these options directly match [tex]\( x^{\frac{11}{14}} \)[/tex].
Thus, the possible choice given the structure might be closest in concept to understanding powers of [tex]\( x\)[/tex]. However, since the specific expression is precise as calculated, ideally none of the options would match precisely to the calculation-derived exponent [tex]\( x^{\frac{11}{14}}\)[/tex].
