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Let's break down the given expression \( \sqrt[5]{6^3} \cdot \sqrt[6]{6^5} \) to find its equivalent form.

### Step 1: Rewrite the roots as fractional exponents
We can rewrite the roots as exponents:
\[
\sqrt[5]{6^3} = 6^{\frac{3}{5}}
\]
\[
\sqrt[6]{6^5} = 6^{\frac{5}{6}}
\]

### Step 2: Multiply the terms
Now, we multiply the two terms with the same base (6). When multiplying with the same base, we add the exponents:
\[
6^{\frac{3}{5}} \cdot 6^{\frac{5}{6}} = 6^{\left(\frac{3}{5} + \frac{5}{6}\right)}
\]

### Step 3: Find the common denominator and simplify
We need to add the fractions \( \frac{3}{5} \) and \( \frac{5}{6} \). The common denominator between 5 and 6 is 30.

\[
\frac{3}{5} = \frac{18}{30}, \quad \frac{5}{6} = \frac{25}{30}
\]
\[
\frac{3}{5} + \frac{5}{6} = \frac{18}{30} + \frac{25}{30} = \frac{43}{30}
\]

### Step 4: Final answer
The expression simplifies to:
\[
6^{\frac{43}{30}}
\]
This is the equivalent form of the given expression.

Would you like more details on any step?

Here are some related questions:
1. How do we simplify expressions with fractional exponents?
2. What is the rule for multiplying terms with the same base?
3. How do we add fractions with different denominators?
4. How can this concept be extended to other bases or exponents?
5. What are the properties of exponents involving roots?

**Tip**: Always remember, when multiplying terms with the same base, add the exponents; when dividing, subtract them.

The expression √[5]{6^3} · √[6]{6^5} is equivalent to

Learn how to simplify the expression √[5]{6^3} · √[6]{6^5} by using fractional exponents and exponent rules. This step-by-step guide covers exponents, roots, and how to handle powers of the same base. Suitable for students in Grades 9-12.

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