Let's break down the expression step by step:

### Expression:
\[ 4\sqrt{81} - 83\sqrt{216} + 155\sqrt{32} + \sqrt{225} \]

1. **Simplify each square root**:
   - \( \sqrt{81} = 9 \), so \( 4\sqrt{81} = 4 \times 9 = 36 \)
   - \( \sqrt{216} = \sqrt{36 \times 6} = \sqrt{36} \times \sqrt{6} = 6\sqrt{6} \), so \( 83\sqrt{216} = 83 \times 6\sqrt{6} = 498\sqrt{6} \)
   - \( \sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2} \), so \( 155\sqrt{32} = 155 \times 4\sqrt{2} = 620\sqrt{2} \)
   - \( \sqrt{225} = 15 \)

### Substituted expression:
\[ 36 - 498\sqrt{6} + 620\sqrt{2} + 15 \]

2. **Combine like terms**:
   - Combine the constants: \( 36 + 15 = 51 \)

### Final Expression:
\[ 51 - 498\sqrt{6} + 620\sqrt{2} \]

This is the simplified form of the expression.

Would you like further details or have any questions about the solution?

### Related Questions:
1. How do you simplify square roots that involve prime factorization?
2. What are the properties of square roots in algebraic expressions?
3. How can you rationalize a denominator involving square roots?
4. What is the process for combining like terms in radical expressions?
5. How do you handle addition and subtraction of radicals with different coefficients?
6. What are the common mistakes to avoid when simplifying square root expressions?
7. Can you explain the concept of a perfect square and how it relates to simplifying square roots?
8. How would the solution change if the coefficients of the square roots were fractions?

**Tip**: When simplifying square roots, always check if the number inside the square root can be factored into a perfect square, as it makes the simplification easier.

Learn how to simplify the expression 4√81 - 83√216 + 155√32 + √225 step-by-step. This problem involves simplification of square roots and combining like terms in radical expressions. Suitable for high school students.

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