https://file.aiquickdraw.com/mathbot/file/17251782042953v5apjzf.jpg

The expression in the image is:

\[
36\sqrt{6 + 2\sqrt{5}} - \sqrt{29 - 12\sqrt{5}}
\]

Let's simplify this expression step by step.

### Step 1: Simplify the square roots
We need to simplify each square root individually, starting with the expressions inside the square roots.

#### Simplifying \( \sqrt{6 + 2\sqrt{5}} \)
We assume it can be written as:

\[
\sqrt{6 + 2\sqrt{5}} = \sqrt{a} + \sqrt{b}
\]

Squaring both sides:

\[
6 + 2\sqrt{5} = a + b + 2\sqrt{ab}
\]

By comparing the rational and irrational parts, we have:

\[
a + b = 6 \quad \text{and} \quad 2\sqrt{ab} = 2\sqrt{5}
\]

From \( 2\sqrt{ab} = 2\sqrt{5} \), we get \( \sqrt{ab} = \sqrt{5} \), so \( ab = 5 \).

Now we solve the system of equations:

\[
a + b = 6, \quad ab = 5
\]

The solutions are the roots of the quadratic equation:

\[
x^2 - 6x + 5 = 0
\]

Solving for \( x \):

\[
x = \frac{6 \pm \sqrt{36 - 20}}{2} = \frac{6 \pm \sqrt{16}}{2} = \frac{6 \pm 4}{2}
\]

So, \( x = 5 \) or \( x = 1 \). Hence, \( a = 5 \) and \( b = 1 \), meaning:

\[
\sqrt{6 + 2\sqrt{5}} = \sqrt{5} + 1
\]

Thus,

\[
36\sqrt{6 + 2\sqrt{5}} = 36(\sqrt{5} + 1) = 36\sqrt{5} + 36
\]

#### Simplifying \( \sqrt{29 - 12\sqrt{5}} \)
Similarly, assume:

\[
\sqrt{29 - 12\sqrt{5}} = \sqrt{a} - \sqrt{b}
\]

Squaring both sides:

\[
29 - 12\sqrt{5} = a + b - 2\sqrt{ab}
\]

By comparing the rational and irrational parts, we have:

\[
a + b = 29 \quad \text{and} \quad -2\sqrt{ab} = -12\sqrt{5}
\]

From \( -2\sqrt{ab} = -12\sqrt{5} \), we get \( \sqrt{ab} = 6\sqrt{5} \), so \( ab = 180 \).

Now, solve the system:

\[
a + b = 29, \quad ab = 180
\]

The quadratic equation is:

\[
x^2 - 29x + 180 = 0
\]

Solving for \( x \):

\[
x = \frac{29 \pm \sqrt{841 - 720}}{2} = \frac{29 \pm \sqrt{121}}{2} = \frac{29 \pm 11}{2}
\]

So, \( x = 20 \) or \( x = 9 \). Hence, \( a = 20 \) and \( b = 9 \), meaning:

\[
\sqrt{29 - 12\sqrt{5}} = \sqrt{20} - \sqrt{9} = 2\sqrt{5} - 3
\]

### Step 2: Substitute back and combine terms
Now, substitute back into the original expression:

\[
36\sqrt{5} + 36 - (2\sqrt{5} - 3)
\]

Distribute the subtraction:

\[
36\sqrt{5} + 36 - 2\sqrt{5} + 3
\]

Combine like terms:

\[
(36\sqrt{5} - 2\sqrt{5}) + (36 + 3) = 34\sqrt{5} + 39
\]

So the simplified expression is:

\[
\boxed{34\sqrt{5} + 39}
\]

Would you like more details or have any questions?

### Related Questions:
1. What are the steps to solving a quadratic equation?
2. How do you simplify square roots involving sums or differences of square roots?
3. Can you explain the binomial expansion used in this context?
4. How can we verify the roots of a quadratic equation?
5. What are some common mistakes to avoid when simplifying expressions involving square roots?

### Tip:
When simplifying expressions involving square roots of binomials, always check if the expression inside the square root can be factored or simplified further using algebraic identities.

Learn how to simplify the expression involving square roots: 36 times the square root of 6 plus 2 times the square root of 5, minus the square root of 29 minus 12 times the square root of 5. Follow step-by-step instructions with detailed explanations and algebraic simplification. Suitable for advanced high school students and beyond.

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