https://r2.erweima.ai/mathbot/image/1727035982344orqvlnc9.jpg

The expression you uploaded is:

\[
\frac{\sqrt{3} \cdot \sqrt[3]{9}}{\sqrt[6]{3}}
\]

Let me walk through simplifying this expression step-by-step:

### Step 1: Express everything as exponents
- The square root of 3 is \( 3^{1/2} \).
- The cube root of 9 is \( 9^{1/3} \).
- The sixth root of 3 is \( 3^{1/6} \).

Thus, the expression becomes:

\[
\frac{3^{1/2} \cdot 9^{1/3}}{3^{1/6}}
\]

### Step 2: Simplify \( 9^{1/3} \)
Recall that \( 9 = 3^2 \), so:

\[
9^{1/3} = (3^2)^{1/3} = 3^{2/3}
\]

Now the expression is:

\[
\frac{3^{1/2} \cdot 3^{2/3}}{3^{1/6}}
\]

### Step 3: Combine exponents in the numerator
When multiplying terms with the same base, add their exponents:

\[
3^{1/2} \cdot 3^{2/3} = 3^{\left(\frac{1}{2} + \frac{2}{3}\right)}
\]

To add these fractions, find a common denominator:

\[
\frac{1}{2} + \frac{2}{3} = \frac{3}{6} + \frac{4}{6} = \frac{7}{6}
\]

So the numerator simplifies to \( 3^{7/6} \). Now the entire expression becomes:

\[
\frac{3^{7/6}}{3^{1/6}}
\]

### Step 4: Subtract exponents in the denominator
When dividing terms with the same base, subtract their exponents:

\[
3^{\frac{7}{6}} \div 3^{\frac{1}{6}} = 3^{\left(\frac{7}{6} - \frac{1}{6}\right)} = 3^{6/6} = 3^1
\]

### Final Simplified Expression:
\[
3^1 = 3
\]

So, the value of the expression is **3**.

Would you like any additional details or explanations?

### Here are 5 related questions to explore this further:
1. How do you simplify expressions with different types of roots (e.g., square roots and cube roots)?
2. What is the general rule for adding and subtracting exponents with the same base?
3. How would you simplify an expression with a higher root, like a 4th root?
4. What are the steps to rationalize the denominator when there are roots involved?
5. How does exponentiation work for fractional powers, and how can you interpret them geometrically?

### Tip:
When working with roots and exponents, converting everything into fractional exponents often makes simplifying the expression easier.

Simplify \\frac{\\sqrt{3} \\cdot \\sqrt[3]{9}}{\\sqrt[6]{3}}

Learn how to simplify the expression \\frac{\\sqrt{3} \\cdot \\sqrt[3]{9}}{\\sqrt[6]{3}}. This step-by-step guide covers key concepts like exponents, roots, and the laws of exponents. Ideal for students in Grades 9-11.

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