https://r2.erweima.ai/mathbot/image/1726447572519xugvrafq.jpg

The expression in the image is:

\[
\sqrt{80} - 2\sqrt{252} + 3\sqrt{405} - 3\sqrt{500}
\]

Let me simplify each square root term:

1. \(\sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5}\)
2. \(\sqrt{252} = \sqrt{4 \times 63} = 2\sqrt{63} = 2\sqrt{9 \times 7} = 6\sqrt{7}\)
3. \(\sqrt{405} = \sqrt{9 \times 45} = 3\sqrt{45} = 3\sqrt{9 \times 5} = 9\sqrt{5}\)
4. \(\sqrt{500} = \sqrt{100 \times 5} = 10\sqrt{5}\)

Now substitute these into the original expression:

\[
4\sqrt{5} - 2(6\sqrt{7}) + 3(9\sqrt{5}) - 3(10\sqrt{5})
\]

Simplify the expression:

\[
4\sqrt{5} - 12\sqrt{7} + 27\sqrt{5} - 30\sqrt{5}
\]

Now combine like terms:

- For \(\sqrt{5}\): \(4\sqrt{5} + 27\sqrt{5} - 30\sqrt{5} = 1\sqrt{5}\)
- For \(\sqrt{7}\): \(-12\sqrt{7}\)

Thus, the simplified expression is:

\[
\sqrt{5} - 12\sqrt{7}
\]

Would you like more details on the simplifications, or do you have any questions?

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Here are 5 related questions to expand on this:

1. How do you simplify square roots with large numbers?
2. Can you simplify \(\sqrt{75} + 2\sqrt{48} - 3\sqrt{27}\)?
3. What is the general method for combining like terms with square roots?
4. How does the distributive property apply to expressions involving square roots?
5. How do we approximate values like \(\sqrt{5}\) and \(\sqrt{7}\) for practical calculations?

**Tip:** When simplifying square roots, always factor out perfect squares to reduce the expression.

Simplify the expression: √80 - 2√252 + 3√405 - 3√500

Learn how to simplify the radical expression √80 - 2√252 + 3√405 - 3√500 using basic algebraic concepts such as factoring perfect squares and combining like terms. This step-by-step guide walks through simplifying square roots and radicals, ideal for students in Grades 9-11.

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