$$ \sqrt{\frac{10\cdot35}{9\cdot27}}=\frac{5}{9}\sqrt{\frac{14}{3}}=\boxed{\frac{5\sqrt{42}}{27}} $$

Let's verify and break down the steps of this expression:

### Given equation:
\[
\sqrt{\frac{10 \cdot 35}{9 \cdot 27}} = \frac{5}{9}\sqrt{\frac{14}{3}} = \boxed{\frac{5\sqrt{42}}{27}}
\]

#### Step 1: Simplify the square root on the left-hand side
The original expression inside the square root is:
\[
\frac{10 \cdot 35}{9 \cdot 27}
\]

Simplify the multiplication in both the numerator and denominator:
\[
10 \cdot 35 = 350, \quad 9 \cdot 27 = 243
\]

Thus, the expression becomes:
\[
\sqrt{\frac{350}{243}}
\]

#### Step 2: Simplify using factors
We can factor out the numbers to further simplify:
\[
350 = 2 \cdot 5^2 \cdot 7, \quad 243 = 3^5
\]

This gives us:
\[
\sqrt{\frac{2 \cdot 5^2 \cdot 7}{3^5}} = \frac{5}{9} \cdot \sqrt{\frac{14}{3}}
\]

#### Step 3: Evaluate the right-hand side
We have:
\[
\frac{5}{9} \sqrt{\frac{14}{3}}
\]

Multiplying by the square root:
\[
\frac{5 \cdot \sqrt{14}}{9 \cdot \sqrt{3}} = \frac{5 \cdot \sqrt{42}}{27}
\]

Thus, the original expression is correct, and the boxed result:
\[
\boxed{\frac{5\sqrt{42}}{27}}
\]
is verified as correct.

Would you like further details on this?

Here are 5 related questions:
1. How can we simplify complex square root expressions further?
2. What are other examples of simplifying fractional square roots?
3. How does the distributive property apply when simplifying square roots?
4. What is the role of prime factorization in simplifying square roots?
5. How can we rationalize denominators in expressions involving square roots?

**Tip:** When simplifying square roots of fractions, it often helps to factor the numbers inside the root to find perfect squares.

Explore how to simplify the square root of a fractional expression, involving prime factorization and fractional multiplication. This problem illustrates the step-by-step process of breaking down square roots of complex fractions, suitable for students in Grades 9-11.

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