Simplify a quotient of monomials using index laws: Evaluate 6a^3 b^3 c^2 ÷ 2a b^2 c.
Aptitude
Surds and Indices
Difficulty: Easy
Choose an option
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A3a^2 b c
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B3a b^2 c
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C3a^2 b^2 c^2
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D3a^3 b^3 c^3
Answer
Correct Answer: 3a^2 b c
Explanation
Introduction / Context:This is a straightforward monomial simplification using exponent subtraction for like bases and coefficient division. Such manipulations are routine in algebraic simplification before factorization or substitution steps.Given Data / Assumptions:
- Expression: (6a^3 b^3 c^2) / (2a b^2 c).
Concept / Approach:For like bases, a^m / a^n = a^(m−n). Also divide numeric coefficients: 6/2 = 3.Step-by-Step Solution:
Coefficient: 6 ÷ 2 = 3.a-exponent: a^(3−1) = a^2.b-exponent: b^(3−2) = b.c-exponent: c^(2−1) = c.Result: 3a^2 b c.Verification / Alternative check:Plug small values (e.g., a=b=c=2): LHS = 6*8*8*4 / (2*2*4*2) = 1536 / 32 = 48; RHS = 3*(4)*2*2 = 48. Matches.
Why Other Options Are Wrong:
- They have incorrect exponent arithmetic or extra factors, contradicting the laws a^m / a^n = a^(m−n).
Common Pitfalls:Adding exponents during division or miscounting one of the variables’ exponents.
Final Answer:
3a^2 b c