To simplify the expression √13 + 30√2 + √(9 + 4√2), we can start by simplifying the square root terms.

√13 is already simplified, so we leave it as is.

For 30√2, we can simplify it further by factoring out the square root of 2:

30√2 = √(900)√2 = √(900 * 2) = √1800

Now, let's simplify the square root term inside the parentheses:

√(9 + 4√2)

To simplify this, we can try to find a perfect square that can be factored out of the expression inside the square root.

Let's assume that the expression can be written as (a + b√2)^2, where a and b are integers.

Expanding (a + b√2)^2, we get:

(a + b√2)^2 = a^2 + 2ab√2 + 2b^2

Comparing this with the expression inside the square root, we can see that a^2 = 9, 2ab√2 = 4√2, and 2b^2 = 0.

From a^2 = 9, we can determine that a = 3 or -3.

From 2ab√2 = 4√2, we can determine that ab = 2.

Since 2b^2 = 0, we can determine that b = 0.

Therefore, the expression inside the square root can be written as (3 + 0√2)^2 = 3^2 = 9.

Now, let's substitute this back into the original expression:

√13 + √1800 + √9

Simplifying further:

√13 + √1800 + 3

Now, let's simplify √1800:

√1800 = √(900 * 2) = √900√2 = 30√2

Substituting this back into the expression:

√13 + 30√2 + 3

Therefore, the simplified expression is √13 + 30√2 + 3.