Rationalize the denominator of the expression. Then simplify your answer.
3/ square root 21  − 3


To rationalize the denominator of the expression \( \frac{3}{\sqrt{21} - 3} \), we need to multiply both the numerator and the denominator by the conjugate of the denominator, \( \sqrt{21} + 3 \). This will eliminate the square root in the denominator.

Let's work through the steps:

### Step 1: Multiply the numerator and the denominator by the conjugate of \( \sqrt{21} - 3 \).

\[
\frac{3}{\sqrt{21} - 3} \times \frac{\sqrt{21} + 3}{\sqrt{21} + 3} = \frac{3(\sqrt{21} + 3)}{(\sqrt{21} - 3)(\sqrt{21} + 3)}
\]

### Step 2: Simplify the denominator.
The denominator is a difference of squares:

\[
(\sqrt{21} - 3)(\sqrt{21} + 3) = (\sqrt{21})^2 - 3^2 = 21 - 9 = 12
\]

### Step 3: Simplify the numerator.

\[
3(\sqrt{21} + 3) = 3\sqrt{21} + 9
\]

### Step 4: Combine the results.

Now the expression becomes:

\[
\frac{3\sqrt{21} + 9}{12}
\]

### Step 5: Simplify the fraction.

We can factor out a 3 from the numerator:

\[
\frac{3(\sqrt{21} + 3)}{12} = \frac{\sqrt{21} + 3}{4}
\]

Thus, the simplified and rationalized form of the expression is:

\[
\boxed{\frac{\sqrt{21} + 3}{4}}
\]

Would you like any further explanation or details on this process? 

Here are five questions to deepen understanding:
1. How do you find the conjugate of an expression?
2. Why does multiplying by the conjugate rationalize the denominator?
3. What is the difference of squares, and how is it used here?
4. Can you apply this rationalization method to another similar expression?
5. How do you factor a common term from a binomial expression?

**Tip:** Always simplify expressions by factoring common terms to make calculations easier!

Rationalize the denominator of the expression. Then simplify your answer. 3/ square root 21 − 3

Learn how to rationalize the denominator of the expression 3 / (√21 - 3) and simplify it step by step. This problem covers algebraic concepts such as rationalization using conjugates and the difference of squares. Suitable for students in Grades 9-10.

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