Let's solve the given expression step by step to show that:

$\sqrt{27} + \sqrt{21} \times \sqrt{7} - \frac{6}{\sqrt{3}} = 8 \sqrt{3}$

Step 1: Simplify $\sqrt{27}$

$\sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3 \sqrt{3}$

Step 2: Simplify $\sqrt{21} \times \sqrt{7}$

$\sqrt{21} \times \sqrt{7} = \sqrt{21 \times 7} = \sqrt{147}$ Next, simplify $\sqrt{147}$: $147 = 49 \times 3 \quad \Rightarrow \quad \sqrt{147} = \sqrt{49 \times 3} = \sqrt{49} \times \sqrt{3} = 7 \sqrt{3}$

Step 3: Simplify $\frac{6}{\sqrt{3}}$

$\frac{6}{\sqrt{3}} = \frac{6}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{6 \sqrt{3}}{3} = 2 \sqrt{3}$

Step 4: Substitute into the original expression

Now, substitute all simplified terms back into the expression: $3 \sqrt{3} + 7 \sqrt{3} - 2 \sqrt{3}$

Step 5: Combine like terms

$(3 + 7 - 2) \sqrt{3} = 8 \sqrt{3}$

This confirms that the left-hand side equals the right-hand side, so:

$\boxed{8 \sqrt{3} = 8 \sqrt{3}}$

Thus, the equation is valid.

Would you like more details on any of the steps?

Here are some related questions:

1. How do you simplify expressions involving square roots?
2. What are the rules for multiplying square roots?
3. How do you rationalize a denominator?
4. What are some other examples of simplifying square root expressions?
5. How can you factor numbers under a square root efficiently?

Tip: Always look for perfect square factors inside square roots to simplify them quickly!