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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!

Show that \(\sqrt{27} + \sqrt{21} \times \sqrt{7} - \frac{6}{\sqrt{3}} = 8\sqrt{3}\)

Learn how to simplify square roots and rationalize the denominator in the expression \(\sqrt{27} + \sqrt{21} \times \sqrt{7} - \frac{6}{\sqrt{3}}\). This step-by-step solution covers key algebraic techniques, suitable for students in Grades 10-12.

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