If a gt 0 and b gt 0, which of the following expressions is equivalent to 10a^5/(sqrt(100a^6b^(12)))?

1. TRANSLATE the problem information

• Given expression: \(\frac{10a^5}{\sqrt{100a^6b^{12}}}\)
• Given constraints: \(a \gt 0\) and \(b \gt 0\) (this means we take positive square roots)

2. INFER the approach

• The key insight is to simplify the denominator first by breaking down the radical
• Once the radical is simplified, we can apply basic division rules for exponents

3. SIMPLIFY the radical in the denominator

• Break apart the radical: \(\sqrt{100a^6b^{12}} = \sqrt{100} \times \sqrt{a^6} \times \sqrt{b^{12}}\)
• Evaluate each piece:
  • \(\sqrt{100} = 10\)
  • \(\sqrt{a^6} = a^3\) (since \(a \gt 0\))
  • \(\sqrt{b^{12}} = b^6\) (since \(b \gt 0\))
• Combined result: \(10a^3b^6\)

4. SIMPLIFY the overall expression

• Now we have: \(\frac{10a^5}{10a^3b^6}\)
• SIMPLIFY by dividing each component:
  • Coefficients: \(10/10 = 1\)
  • For variable a: \(a^5/a^3 = a^{(5-3)} = a^2\)
  • For variable b: \(1/b^6 = b^{-6}\)

Answer: \(a^2b^{-6}\), which is Choice C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution with negative exponents: Students correctly simplify most of the expression but write \(1/b^6\) as \(b^6\) instead of \(b^{-6}\), forgetting that division by a term creates a negative exponent.

This reasoning leads them to get \(a^2b^6\), causing them to select Choice D (\(a^2b^6\)).

Second Most Common Error:

SIMPLIFY error in exponent subtraction: Students make an error when computing \(a^5/a^3\), either thinking it equals \(a^{-1}\) (subtracting backwards: \(3-5\)) or leaving it as \(a^3\) (not recognizing the division rule).

The backwards subtraction error leads them to select Choice A (\(a^{-1}b^{-6}\)), while the division confusion leads to Choice B (\(a^3b^{-6}\)).

The Bottom Line:

This problem tests systematic algebraic manipulation skills - students must correctly handle both radical simplification and multiple exponent rules in sequence. The negative exponent notation is particularly tricky since it's easy to forget that "one over something" becomes a negative exponent.