Which of the following is the simplified form of $\sqrt{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x}$ ? $x^{\frac{3}{7}}$ $x^{\frac{1}{7}}$ $x^{\frac{3}{21}}$ $21 \sqrt{x}$
Solution
Step 1 ：The given expression is \(\sqrt{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x}\). We know that \(\sqrt{x}\) is equivalent to \(x^{\frac{1}{2}}\) and \(\sqrt[7]{x}\) is equivalent to \(x^{\frac{1}{7}}\). So, the expression can be rewritten as \(x^{\frac{1}{2}} \cdot x^{\frac{1}{7}} \cdot x^{\frac{1}{7}}\).
Step 2 ：When multiplying expressions with the same base, we add the exponents. So, the simplified form of the expression is \(x^{\frac{1}{2} + \frac{1}{7} + \frac{1}{7}}\).
Step 3 ：The simplified form of the expression is \(x^{0.785714285714286}\). However, this is not in the form of a fraction. We need to convert this decimal to a fraction to match the answer choices.
Step 4 ：The decimal 0.785714285714286 can be converted to the fraction \(\frac{11}{14}\).
Step 5 ：\(\boxed{x^{\frac{11}{14}}}\) is the simplified form of the expression \(\sqrt{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x}\). However, this option is not given in the answer choices. There might be a mistake in the question or the answer choices.
