Which expression is equivalent to \((8\mathrm{y}\mathrm{z})(\mathrm{y})(7\mathrm{z})\)?

1. TRANSLATE the problem information

• Given: The expression \((8\mathrm{yz})(\mathrm{y})(7\mathrm{z})\)
• Need to find: An equivalent simplified expression

2. INFER the approach

• This is a multiplication problem with algebraic terms
• Strategy: Group like terms together (coefficients with coefficients, same variables with same variables)
• This will let us use exponent rules to simplify

3. SIMPLIFY by regrouping terms

Rewrite \((8\mathrm{yz})(\mathrm{y})(7\mathrm{z})\) as: \(8 \times \mathrm{y} \times \mathrm{z} \times \mathrm{y} \times 7 \times \mathrm{z}\)

Use the commutative property to rearrange:
\((8 \times 7) \times (\mathrm{y} \times \mathrm{y}) \times (\mathrm{z} \times \mathrm{z})\)

4. SIMPLIFY each group separately

• Coefficients: \(8 \times 7 = 56\)
• y terms: \(\mathrm{y} \times \mathrm{y} = \mathrm{y}^1 \times \mathrm{y}^1 = \mathrm{y}^{(1+1)} = \mathrm{y}^2\)
• z terms: \(\mathrm{z} \times \mathrm{z} = \mathrm{z}^1 \times \mathrm{z}^1 = \mathrm{z}^{(1+1)} = \mathrm{z}^2\)

5. SIMPLIFY to get the final answer

Combine: \(56 \times \mathrm{y}^2 \times \mathrm{z}^2 = 56\mathrm{y}^2\mathrm{z}^2\)

Answer: A. \(56\mathrm{y}^2\mathrm{z}^2\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students forget to multiply all terms or incorrectly apply exponent rules

Many students might multiply \((8\mathrm{yz})\) by \((\mathrm{y})\) correctly to get \(8\mathrm{y}^2\mathrm{z}\), but then forget to multiply by the final \((7\mathrm{z})\) term, or they multiply by 7 but forget about the z. Others know they need to combine like variables but incorrectly think \(\mathrm{y} \times \mathrm{y} = 2\mathrm{y}\) instead of \(\mathrm{y}^2\).

This may lead them to select Choice B (\(56\mathrm{y}^2\mathrm{z}\)) or Choice C (\(56\mathrm{yz}\))

Second Most Common Error:

Conceptual confusion about coefficient multiplication: Students add coefficients instead of multiplying them

Some students see 8 and 7 and think "\(8 + 7 = 15\)" instead of "\(8 \times 7 = 56\)". Combined with forgetting exponent rules, this leads to answers like 15yz or 16yz.

This may lead them to select Choice D (\(16\mathrm{yz}\))

The Bottom Line:

This problem tests whether students can systematically apply multiplication rules to algebraic expressions. Success requires both proper regrouping strategy AND accurate execution of coefficient multiplication and exponent rules.