Simplify the radical.
StartRoot left parenthesis negative 27 right parenthesis squared plus left parenthesis negative 9 right parenthesis squared EndRoot

Let's simplify the given radical expression:

\[
\sqrt{(-27)^2 + (-9)^2}
\]

### Step 1: Square the terms inside the radical.
\[
(-27)^2 = 729
\]
\[
(-9)^2 = 81
\]

### Step 2: Add the squared terms.
\[
729 + 81 = 810
\]

### Step 3: Take the square root of 810.
\[
\sqrt{810}
\]

### Step 4: Simplify \(\sqrt{810}\).
First, factor 810:
\[
810 = 2 \times 3^4 \times 5
\]

Now, we simplify the square root:
\[
\sqrt{810} = \sqrt{2 \times 3^4 \times 5} = 3^2 \times \sqrt{2 \times 5} = 9\sqrt{10}
\]

Thus, the simplified form is:
\[
9\sqrt{10}
\]

Would you like further clarification on any of the steps?

Here are 5 related questions:
1. How do you simplify a square root with a non-perfect square inside?
2. What is the process for factoring numbers for square roots?
3. How do negative numbers inside squares affect simplification?
4. What is the prime factorization of a number, and why is it useful in simplifying radicals?
5. How can you estimate the value of \(\sqrt{10}\) for an approximate answer?

**Tip**: When simplifying square roots, always look for perfect square factors to make the process easier.

Simplify the radical. StartRoot left parenthesis negative 27 right parenthesis squared plus left parenthesis negative 9 right parenthesis squared EndRoot

Learn how to simplify the radical expression \(\sqrt{(-27)^2 + (-9)^2}\) with a step-by-step solution. This problem involves squaring negative numbers, adding results, and simplifying square roots using prime factorization. Perfect for students in Grades 9-10.

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