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The expression given is:

\[
\sqrt{49 - 12\sqrt{5}}
\]

Let's evaluate this carefully.

### Step 1: Break down the expression inside the square root.

We start with the part inside the square root:

\[
49 - 12\sqrt{5}
\]

Notice that \(49\) is a perfect square, which could hint that the expression under the square root might factor into something simple.

### Step 2: Check for potential factorization or simplification.

We aim to simplify \( 49 - 12\sqrt{5} \) and check if it can be expressed in the form \( (a - b\sqrt{5})^2 \). Let’s expand \( (a - b\sqrt{5})^2 \):

\[
(a - b\sqrt{5})^2 = a^2 - 2ab\sqrt{5} + 5b^2
\]

We want to compare this with \( 49 - 12\sqrt{5} \).

- Matching the terms without square roots: \( a^2 + 5b^2 = 49 \)
- Matching the terms with square roots: \( -2ab\sqrt{5} = -12\sqrt{5} \)

### Step 3: Solve for \(a\) and \(b\).

From \( -2ab = -12 \), we get:

\[
ab = 6
\]

Now, using \( a^2 + 5b^2 = 49 \), we can solve this system of equations.

- From \( ab = 6 \), we have \( a = \frac{6}{b} \).
- Substitute into \( a^2 + 5b^2 = 49 \):

\[
\left(\frac{6}{b}\right)^2 + 5b^2 = 49
\]
\[
\frac{36}{b^2} + 5b^2 = 49
\]

Multiply through by \( b^2 \) to clear the fraction:

\[
36 + 5b^4 = 49b^2
\]

Rearranging this gives:

\[
5b^4 - 49b^2 + 36 = 0
\]

Let \( x = b^2 \), so the equation becomes a quadratic in \(x\):

\[
5x^2 - 49x + 36 = 0
\]

### Step 4: Solve the quadratic equation.

Using the quadratic formula:

\[
x = \frac{-(-49) \pm \sqrt{(-49)^2 - 4(5)(36)}}{2(5)}
\]
\[
x = \frac{49 \pm \sqrt{2401 - 720}}{10}
\]
\[
x = \frac{49 \pm \sqrt{1681}}{10}
\]
\[
x = \frac{49 \pm 41}{10}
\]

So, \( x = \frac{49 + 41}{10} = 9 \) or \( x = \frac{49 - 41}{10} = 0.8 \).

Thus, \( b^2 = 9 \), meaning \( b = 3 \), or \( b^2 = 0.8 \), but this doesn’t give us a simple integer solution.

### Step 5: Calculate \(a\).

If \( b = 3 \), then \( a = \frac{6}{3} = 2 \).

### Step 6: Verify the factorization.

Now, substitute \(a = 2\) and \(b = 3\) into \( (a - b\sqrt{5})^2 \):

\[
(2 - 3\sqrt{5})^2 = 2^2 - 2(2)(3)\sqrt{5} + 5(3)^2
\]
\[
= 4 - 12\sqrt{5} + 45
\]
\[
= 49 - 12\sqrt{5}
\]

This matches the original expression. Therefore:

\[
\sqrt{49 - 12\sqrt{5}} = 2 - 3\sqrt{5}
\]

### Final Answer:

\[
\boxed{2 - 3\sqrt{5}}
\]

Would you like a deeper breakdown of any step? Here are a few related questions you might consider:

1. How do you factor quadratic expressions involving radicals?
2. What is the quadratic formula and how can it be applied to different types of equations?
3. How can we recognize perfect square patterns in expressions?
4. Why is factoring important when simplifying radicals?
5. Can you apply the method used here to similar problems with different numbers?

**Tip:** When simplifying radicals, always check if the expression inside the square root can be factored or rewritten as a perfect square.

What is the value of √(49 - 12√5)? Be very careful – you might get the wrong answer!

This detailed solution explains how to simplify the expression √(49 - 12√5) using factoring techniques and the quadratic formula. Suitable for high school students, this problem covers algebraic concepts such as binomial expansion and radical simplification.

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