7\sqrt{a\ }-\sqrt{8a}+8\sqrt{9a}-\sqrt{32a}

Let's simplify the given expression step by step:

The expression is:

\[
7\sqrt{a} - \sqrt{8a} + 8\sqrt{9a} - \sqrt{32a}
\]

### Step 1: Simplify each square root term.

- \( 7\sqrt{a} \) remains as is.
- \( \sqrt{8a} = \sqrt{4 \times 2a} = \sqrt{4} \times \sqrt{2a} = 2\sqrt{2a} \).
- \( 8\sqrt{9a} = 8 \times \sqrt{9} \times \sqrt{a} = 8 \times 3 \times \sqrt{a} = 24\sqrt{a} \).
- \( \sqrt{32a} = \sqrt{16 \times 2a} = \sqrt{16} \times \sqrt{2a} = 4\sqrt{2a} \).

### Step 2: Substitute the simplified terms back into the expression.

\[
7\sqrt{a} - 2\sqrt{2a} + 24\sqrt{a} - 4\sqrt{2a}
\]

### Step 3: Combine like terms.

- Combine \( 7\sqrt{a} \) and \( 24\sqrt{a} \):

\[
7\sqrt{a} + 24\sqrt{a} = 31\sqrt{a}
\]

- Combine \( -2\sqrt{2a} \) and \( -4\sqrt{2a} \):

\[
-2\sqrt{2a} - 4\sqrt{2a} = -6\sqrt{2a}
\]

### Final Answer:

The simplified expression is:

\[
31\sqrt{a} - 6\sqrt{2a}
\]

Would you like more details or explanations on any of the steps?

Here are 5 related questions to deepen understanding:
1. How do you simplify a square root expression like \( \sqrt{8a} \)?
2. What is the general method to factor out square roots?
3. Can you identify other cases where simplifying radicals leads to similar expressions?
4. How does the distributive property apply when combining square roots?
5. What are some real-world applications of expressions involving square roots?

**Tip:** Always look for perfect squares inside square root terms to simplify expressions efficiently.

7\sqrt{a} - \sqrt{8a} + 8\sqrt{9a} - \sqrt{32a}

This detailed step-by-step solution simplifies the radical expression 7√a - √8a + 8√9a - √32a. It covers algebraic concepts such as combining like terms and simplifying square roots, suitable for students in Grades 9-10.

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