Math Problem Statement

-√(81a^16b^20c^12)

Solution

You provided an expression:

81a16b20c12-\sqrt{81a^{16}b^{20}c^{12}}

Let's simplify it step by step.

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

81=9\sqrt{81} = 9

  1. For the variables under the square root:
  • a16=a8\sqrt{a^{16}} = a^{8} because a16=a16/2\sqrt{a^{16}} = a^{16/2}.
  • b20=b10\sqrt{b^{20}} = b^{10} because b20=b20/2\sqrt{b^{20}} = b^{20/2}.
  • c12=c6\sqrt{c^{12}} = c^{6} because c12=c12/2\sqrt{c^{12}} = c^{12/2}.
  1. Now, apply these simplifications to the original expression:

81a16b20c12=9a8b10c6-\sqrt{81a^{16}b^{20}c^{12}} = -9a^8b^{10}c^6

The fully simplified expression is:

9a8b10c6-9a^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 64a10b6c8\sqrt{64a^{10}b^{6}c^{8}}?
  2. What is the general rule for simplifying xn\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!

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Math Problem Analysis

Mathematical Concepts

Simplification of Radical Expressions
Exponents
Square Roots

Formulas

√(x^n) = x^(n/2)
√(ab) = √a * √b

Theorems

Laws of Exponents

Suitable Grade Level

Grades 9-12