To simplify the expression [tex]\(\sqrt{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x}\)[/tex], let's rewrite each part using exponent notation. The expressions [tex]\(\sqrt{x}\)[/tex], [tex]\(\sqrt[7]{x}\)[/tex], and [tex]\(\sqrt[7]{x}\)[/tex] can be represented as:
- [tex]\(\sqrt{x} = x^{\frac{1}{2}}\)[/tex]
- [tex]\(\sqrt[7]{x} = x^{\frac{1}{7}}\)[/tex]
- [tex]\(\sqrt[7]{x} = x^{\frac{1}{7}}\)[/tex]
Now, we can combine these using the properties of exponents. Specifically, the property that [tex]\(a^{m} \cdot a^{n} = a^{m+n}\)[/tex]:
[tex]\[
\sqrt{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x} = x^{\frac{1}{2}} \cdot x^{\frac{1}{7}} \cdot x^{\frac{1}{7}}
\][/tex]
When we multiply exponents with the same base, we add the exponents together:
[tex]\[
x^{\frac{1}{2}} \cdot x^{\frac{1}{7}} \cdot x^{\frac{1}{7}} = x^{\frac{1}{2} + \frac{1}{7} + \frac{1}{7}}
\][/tex]
To add these fractions, we need a common denominator. The fractions [tex]\(\frac{1}{2}\)[/tex] and [tex]\(\frac{1}{7}\)[/tex] can be converted as follows:
[tex]\[
\frac{1}{2} = \frac{7}{14}
\][/tex]
[tex]\[
\frac{1}{7} = \frac{2}{14}
\][/tex]
Now adding these:
[tex]\[
\frac{1}{2} + \frac{1}{7} + \frac{1}{7} = \frac{7}{14} + \frac{2}{14} + \frac{2}{14} = \frac{7 + 2 + 2}{14} = \frac{11}{14}
\][/tex]
So, we have:
[tex]\[
x^{\frac{1}{2} + \frac{1}{7} + \frac{1}{7}} = x^{\frac{11}{14}}
\][/tex]
From the numerical value, [tex]\(\frac{11}{14}\)[/tex] is approximately equal to [tex]\(0.785714285714286\)[/tex]. Therefore, the simplified form of the given expression is [tex]\(x^{\frac{11}{14}}\)[/tex].
Among the provided options, the correct simplified form is not explicitly listed. Therefore, based on the closest match in fractional form, the simplified form [tex]\(x^{\frac{11}{14}}\)[/tex] corresponds to [tex]\(0.785714285714286\)[/tex].
Hence, the simplified form of [tex]\(\sqrt{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x}\)[/tex] is:
[tex]\[
x^{\frac{11}{14}}
\][/tex]
And since none of the provided options match this, the correct choice should fundamentally be provided as [tex]\(x^{\frac{11}{14}}\)[/tex].
