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You provided an expression:

\[
-\sqrt{81a^{16}b^{20}c^{12}}
\]

Let's simplify it step by step.

1. Start by simplifying the square root of each term.

\[
\sqrt{81} = 9
\]

2. For the variables under the square root:
- \(\sqrt{a^{16}} = a^{8}\) because \(\sqrt{a^{16}} = a^{16/2}\).
- \(\sqrt{b^{20}} = b^{10}\) because \(\sqrt{b^{20}} = b^{20/2}\).
- \(\sqrt{c^{12}} = c^{6}\) because \(\sqrt{c^{12}} = c^{12/2}\).

3. Now, apply these simplifications to the original expression:

\[
-\sqrt{81a^{16}b^{20}c^{12}} = -9a^8b^{10}c^6
\]

The fully simplified expression is:

\[
-9a^8b^{10}c^6
\]

Would you like more details or have any further questions?

Here are some related questions to deepen your understanding:

1. How would you simplify \(\sqrt{64a^{10}b^{6}c^{8}}\)?
2. What is the general rule for simplifying \(\sqrt{x^n}\)?
3. How does the presence of a negative sign outside the square root affect the result?
4. How can we handle fractional exponents in similar problems?
5. Can we simplify cube roots in the same way as square roots?

**Tip:** Always check if coefficients and exponents are perfect squares when simplifying square roots!

Simplify the expression -√(81a^16b^20c^12) step by step by applying square root rules and exponent laws. Learn the process of simplifying large exponents and coefficients, suitable for high school students.

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