The expression in the image is:

$3 \cdot \sqrt{a^5 b^7} \cdot 2 \cdot \sqrt{5 a^7 b^3}$

Let's simplify it step by step.

Step 1: Combine constants

First, combine the constants outside the square roots: $3 \cdot 2 = 6$

Now the expression looks like this: $6 \cdot \sqrt{a^5 b^7} \cdot \sqrt{5 a^7 b^3}$

Step 2: Combine the square roots

Since both terms are under square roots, we can combine them into a single square root: $6 \cdot \sqrt{(a^5 b^7) \cdot (5 a^7 b^3)}$

Step 3: Multiply inside the square root

Now, multiply the terms inside the square root: $(a^5 \cdot a^7) \cdot (b^7 \cdot b^3) \cdot 5 = a^{5+7} \cdot b^{7+3} \cdot 5 = a^{12} \cdot b^{10} \cdot 5$

Now, the expression is: $6 \cdot \sqrt{5 a^{12} b^{10}}$

Step 4: Simplify the square root

We can now simplify the square root: $\sqrt{5 a^{12} b^{10}} = \sqrt{5} \cdot \sqrt{a^{12}} \cdot \sqrt{b^{10}}$ $= \sqrt{5} \cdot a^6 \cdot b^5$

Step 5: Final result

Now, substitute this back into the expression: $6 \cdot \sqrt{5} \cdot a^6 \cdot b^5$

Thus, the simplified expression is: $6 a^6 b^5 \sqrt{5}$

Would you like any further explanation or have any questions on this?

Here are 5 related questions:

1. How do you simplify square roots with variables?
2. What are the properties of square roots when multiplying terms inside the root?
3. Can you simplify an expression with exponents inside and outside square roots?
4. How do you combine square roots with different bases or constants?
5. How do exponents behave when multiplying terms with the same base?

Tip: When simplifying square roots, always check for perfect squares or powers of 2, as these can be simplified further.