square root of x times third root of x times x^1/6

To simplify the expression \(\sqrt{x} \times \sqrt[3]{x} \times x^{1/6}\), let's rewrite each term using exponents and then combine them.

1. **Rewrite the roots as exponents:**

   - The square root of \(x\) is \(\sqrt{x} = x^{1/2}\).
   - The cube root of \(x\) is \(\sqrt[3]{x} = x^{1/3}\).
   - The last term is already in exponent form: \(x^{1/6}\).

   Now the expression is:
   \[
   x^{1/2} \times x^{1/3} \times x^{1/6}
   \]

2. **Combine the exponents:**

   When multiplying terms with the same base, you add the exponents:
   \[
   x^{1/2 + 1/3 + 1/6}
   \]

3. **Find a common denominator and add the exponents:**

   - The common denominator for \(\frac{1}{2}\), \(\frac{1}{3}\), and \(\frac{1}{6}\) is 6.
   - Convert each exponent:
     \[
     \frac{1}{2} = \frac{3}{6}, \quad \frac{1}{3} = \frac{2}{6}, \quad \frac{1}{6} = \frac{1}{6}
     \]
   - Add the fractions:
     \[
     \frac{3}{6} + \frac{2}{6} + \frac{1}{6} = \frac{6}{6} = 1
     \]

   So, the expression simplifies to:
   \[
   x^1 = x
   \]

Thus, the simplified expression is:
\[
\boxed{x}
\]

Would you like more details or have any questions?

Here are some related questions for further practice:

1. Simplify \(\sqrt[4]{x} \times \sqrt[5]{x} \times x^{1/20}\).
2. Simplify \(\sqrt[6]{x^2} \times \sqrt{x^3}\).
3. What is \(\sqrt{y} \times \sqrt[3]{y^2} \times y^{1/2}\)?
4. Simplify \(\sqrt[3]{x^5} \times x^{1/6}\).
5. What is \(\sqrt{x^2} \times \sqrt[3]{x^4} \times x^{1/12}\)?
6. Simplify \(\sqrt[5]{x^3} \times \sqrt[2]{x}\).
7. Simplify \(x^{3/4} \times x^{1/4}\).
8. What is the value of \(\sqrt[3]{y^4} \times \sqrt[2]{y}\)?

**Tip:** When multiplying terms with the same base, always add the exponents to simplify the expression.

Learn how to simplify the expression \(\sqrt{x} \times \sqrt[3]{x} \times x^{1/6}\) step-by-step. This problem covers key mathematical concepts including exponents and roots. Suitable for students in Grades 7-9.

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