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Let's go through the given expression step by step.

We are asked to show that:
\[
\sqrt{\frac{85}{21}} \times \sqrt{\frac{63}{136}} = \frac{1}{4} \times \sqrt{30}
\]

### Step 1: Combine the square roots

By using the property of square roots that \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\), we can combine the two square roots on the left-hand side:

\[
\sqrt{\frac{85}{21}} \times \sqrt{\frac{63}{136}} = \sqrt{\frac{85 \times 63}{21 \times 136}}
\]

### Step 2: Simplify the numerator and denominator

Now, we calculate the numerator and the denominator.

- The numerator: \(85 \times 63 = 5355\)
- The denominator: \(21 \times 136 = 2856\)

So, we have:

\[
\sqrt{\frac{5355}{2856}}
\]

### Step 3: Simplify the fraction

We now simplify the fraction \(\frac{5355}{2856}\). Let's find the greatest common divisor (GCD) of 5355 and 2856. 

- The GCD of 5355 and 2856 is 178.

Dividing both the numerator and denominator by 178:

\[
\frac{5355}{2856} = \frac{5355 \div 178}{2856 \div 178} = \frac{30}{16}
\]

Thus, the expression becomes:

\[
\sqrt{\frac{30}{16}} = \frac{\sqrt{30}}{\sqrt{16}} = \frac{\sqrt{30}}{4}
\]

### Step 4: Compare with the right-hand side

The right-hand side of the equation is \(\frac{1}{4} \times \sqrt{30}\), which is exactly the same as \(\frac{\sqrt{30}}{4}\).

Thus, both sides are equal, and we have shown that:

\[
\sqrt{\frac{85}{21}} \times \sqrt{\frac{63}{136}} = \frac{1}{4} \times \sqrt{30}
\]

Would you like any further clarification on this? Here are five questions that could extend your understanding:

1. How do you determine the greatest common divisor (GCD) of two numbers?
2. Why can we combine square roots in multiplication?
3. What other properties of square roots are useful in solving similar problems?
4. How does simplifying fractions help in solving equations involving square roots?
5. How do you calculate the square root of a fraction?

**Tip:** When multiplying square roots of fractions, it can be easier to combine the fractions first, then take the square root of the result to avoid unnecessary calculations.

Show that: \(\sqrt{\frac{85}{21}} \times \sqrt{\frac{63}{136}} = \frac{1}{4} \times \sqrt{30}\)

Learn how to simplify the expression \(\sqrt{\frac{85}{21}} \times \sqrt{\frac{63}{136}}\) and show it equals \(\frac{1}{4} \times \sqrt{30}\). This problem covers properties of square roots, fraction multiplication, and simplification. Suitable for students in Grades 9-11.

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