Simplify (ab^2 c^3) * (a^4 b^5 c^-6) / (a^-7 b^-8 c^9) * (a^10 b^11 c^-12) * (a^13 b^-14 c^15) 

Simplify (ab^2 c^3) * (a^4 b^5 c^-6) / (a^-7 b^-8 c^9) * (a^10 b^11 c^-12) * (a^13 b^-14 c^15)

While simplifying (ab^2 c^3) * (a^4 b^5 c^-6) / (a^-7 b^-8 c^9) * (a^10 b^11 c^-12) * (a^13 b^-14 c^15) is a^-11b^18c^-15

Explanation

Here's a summary of the reasoning:

1. Combining exponents: We combined exponents within each parenthesis and across multiplied terms, resulting in a simplified expression with terms like a^5b^7c^-3and a^16b^-11c^12.
2. Canceling common terms: We identified and canceled common terms appearing in both the numerator and denominator, reducing the expression further.
3. Simplifying exponents again: We combined remaining exponents where possible, like b^15b^11 to b^26, and then simplified further by canceling c^15 and c^12.
4. Final result: This process led to the final simplified expression a^-11b^18c^-15.

Therefore, based on the provided steps and considering the initial expression, a^-11b^18c^-15 is the correct and complete answer.

Algebraic Expression and Simplification

Here's a comprehensive explanation of algebraic expressions and simplification:

Algebraic Expressions:

• Definition: A combination of variables, numbers, and operations (like addition, subtraction, multiplication, division, and exponentiation) that represents a mathematical relationship.
• Examples:
  • 3x + 5y - 2
  • 2a^ - 4ab + b^3
  • (x + y) / (2z)
  • √(x^2 + y^2)

Simplification of Algebraic Expressions:

Goal: To write the expression in a more concise and manageable form without changing its value.

Key Steps:

1. Combine Like Terms:
   • Identify terms that have the same variables and exponents.
   • Add or subtract their coefficients (numerical factors).
   • Example: 5x^2 + 3x^2 - 4x^2 = 4x^2
2. Apply the Distributive Property:
   • Multiply any factors outside of parentheses with those inside.
   • Example: 2(x + 3) = 2x + 6
3. Remove Parentheses:
   • If no multiplication is involved, simply remove them.
   • Example: (a + b) - c = a + b - c
4. Simplify Exponents:
   • Use rules of exponents to combine terms with the same base.
   • Example: a^3 * ^2^2 = a^5