Which expression is equivalent to \(5(\mathrm{a}^6 - 64\mathrm{b}^2)\)?

1. INFER the factoring pattern

Looking at \(\mathrm{5(a^6 - 64b^2)}\), I need to factor the expression inside the parentheses first.

• I notice this has the form "something squared minus something else squared"
• \(\mathrm{a^6 = (a^3)^2}\) and \(\mathrm{64b^2 = (8b)^2}\)
• This means \(\mathrm{a^6 - 64b^2 = (a^3)^2 - (8b)^2}\)

This is a difference of squares pattern!

2. SIMPLIFY using the difference of squares formula

• The difference of squares formula is: \(\mathrm{x^2 - y^2 = (x - y)(x + y)}\)
• Here, \(\mathrm{x = a^3}\) and \(\mathrm{y = 8b}\)
• So \(\mathrm{(a^3)^2 - (8b)^2 = (a^3 - 8b)(a^3 + 8b)}\)

3. Include the coefficient

• Don't forget the 5 in front: \(\mathrm{5(a^6 - 64b^2) = 5(a^3 - 8b)(a^3 + 8b)}\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that \(\mathrm{a^6}\) and \(\mathrm{64b^2}\) are both perfect squares that can be written as \(\mathrm{(a^3)^2}\) and \(\mathrm{(8b)^2}\) respectively.

Instead, they might try to factor this as if it were a perfect square trinomial, looking for something like \(\mathrm{(a^3 - 8b)^2}\) and selecting Choice A (\(\mathrm{5(a^3 - 8b)(a^3 - 8b)}\)).

Second Most Common Error:

Conceptual confusion about exponent relationships: Students might think \(\mathrm{a^6}\) factors as \(\mathrm{(a^6)^1}\) rather than recognizing it equals \(\mathrm{(a^3)^2}\).

This leads them to incorrectly apply the difference of squares formula with the wrong terms, potentially selecting Choice C (\(\mathrm{5(a^6 - 8b)(a^6 + 8b)}\)).

The Bottom Line:

The key insight is recognizing that both \(\mathrm{a^6}\) and \(\mathrm{64b^2}\) are perfect squares that can be rewritten to reveal the difference of squares pattern. Without this recognition, students can't access the correct factoring approach.