To simplify the given expression [tex]\(\frac{x^{\frac{4}{7}} \cdot x^{\frac{3}{7}}}{x^{\frac{1}{7}}}\)[/tex], follow these steps:
1. Combine the exponents in the numerator:
[tex]\[
x^{\frac{4}{7}} \cdot x^{\frac{3}{7}} = x^{\left(\frac{4}{7} + \frac{3}{7}\right)}
\][/tex]
Since the bases are the same (both [tex]\(x\)[/tex]), you can add the exponents:
[tex]\[
x^{\frac{4}{7} + \frac{3}{7}} = x^{\frac{7}{7}} = x^1 = x
\][/tex]
2. Simplify the fraction with the consolidated exponent:
[tex]\[
\frac{x}{x^{\frac{1}{7}}}
\][/tex]
This can be rewritten as:
[tex]\[
x^{1} \div x^{\frac{1}{7}}
\][/tex]
When dividing like bases, subtract the exponents:
[tex]\[
x^{1 - \frac{1}{7}} = x^{\frac{7}{7} - \frac{1}{7}} = x^{\frac{6}{7}}
\][/tex]
Therefore, the simplified expression is:
[tex]\[ x^{\frac{6}{7}} \][/tex]
The correct answer is: [tex]\(x^{\frac{6}{7}}\)[/tex].
