Math Problem Statement

Simplify the expression \(\frac{(-2a^{-2}b^3c^{-3})^2}{(-3a^{-4}b^{-2}c^{-2})^2}\).

Solution

Let's simplify the given expression step by step.

Given expression:

(2a2b3c3)2(3a4b2c2)2\frac{(-2a^{-2}b^3c^{-3})^2}{(-3a^{-4}b^{-2}c^{-2})^2}

Step 1: Apply the power rule (xy)m=xmym(xy)^m = x^m y^m

Each term inside the parentheses is squared:

Numerator: (2)2(a2)2(b3)2(c3)2(-2)^2 (a^{-2})^2 (b^3)^2 (c^{-3})^2

4a4b6c64a^{-4}b^6c^{-6}

Denominator: (3)2(a4)2(b2)2(c2)2(-3)^2 (a^{-4})^2 (b^{-2})^2 (c^{-2})^2

9a8b4c49a^{-8}b^{-4}c^{-4}

Thus, the expression becomes:

4a4b6c69a8b4c4\frac{4a^{-4}b^6c^{-6}}{9a^{-8}b^{-4}c^{-4}}

Step 2: Simplify each base separately using xmxn=xmn\frac{x^m}{x^n} = x^{m-n}

  • For aa:
    a4(8)=a4a^{-4 - (-8)} = a^{4}

  • For bb:
    b6(4)=b10b^{6 - (-4)} = b^{10}

  • For cc:
    c6(4)=c2c^{-6 - (-4)} = c^{-2}

Thus, we get:

49a4b10c2\frac{4}{9} a^4 b^{10} c^{-2}

Step 3: Rewrite the final answer

Since c2c^{-2} is in the numerator, move it to the denominator:

4a4b109c2\frac{4a^4b^{10}}{9c^2}

Final Answer:

4a4b109c2\frac{4a^4b^{10}}{9c^2}

Would you like a more detailed breakdown of any step?

Here are five related questions to deepen your understanding:

  1. How do you simplify expressions with negative exponents?
  2. What is the general rule for distributing exponents over multiplication?
  3. How would the solution change if different exponents were used?
  4. What happens when the base inside parentheses is negative?
  5. How does exponentiation affect fraction simplification?

Tip: Always simplify exponents step by step and apply exponent rules carefully to avoid mistakes! 😊

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Math Problem Analysis

Mathematical Concepts

Algebra
Exponent Rules
Simplifying Expressions

Formulas

Power of a product rule: \((xy)^m = x^m y^m\)
Quotient of powers rule: \(\frac{x^m}{x^n} = x^{m-n}\)
Negative exponent rule: \(x^{-n} = \frac{1}{x^n}\)

Theorems

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Suitable Grade Level

Grades 8-10