Questions: Simplifying Radical Expressions

Question 1

A student simplifies √72 as follows: √72 = √(4 × 18) = 2√18. Is the expression fully simplified?

• A. Yes — 4 is a perfect square factor, so the simplification is complete
• B. No — √18 still contains a perfect square factor (9), so the answer should be 6√2
• C. No — the student should have left the answer as √72
• D. Yes — the product rule only needs to be applied once

Answer: (B)

The expression is not fully simplified because √18 = √(9 × 2) = 3√2, so 2√18 = 6√2. A radical is fully simplified only when no perfect square factors remain under the radical sign. This is why finding the largest perfect square factor (36 in this case, not 4) saves steps: √72 = √(36 × 2) = 6√2 in one step instead of two.

Question 2

Which of the following correctly applies the product rule for radicals?

• A. √(9 + 16) = √9 + √16 = 3 + 4 = 7
• B. √(9 × 16) = √9 × √16 = 3 × 4 = 12
• C. √(9 + 16) = √9 × √16 = 12
• D. √(9 × 16) = √9 + √16 = 7

Answer: (B)

The product rule states √(ab) = √a · √b — it applies to multiplication, not addition. √(9 × 16) = √9 × √16 = 3 × 4 = 12, which is correct (since √144 = 12). The first option is the most common error: applying a sum rule that does not exist. √(9 + 16) = √25 = 5, not 7.

Question 3

√(9 + 16) = √9 + √16 = 7

• T. True
• F. False

Answer: False

The product rule for radicals (√(ab) = √a · √b) does NOT extend to sums. √(9 + 16) = √25 = 5, while √9 + √16 = 3 + 4 = 7. These are different values. The 'sum rule for radicals' is a fiction that trips up students from algebra through calculus — the product rule is the only valid splitting operation.

Question 4

√(4 × 25) = √4 × √25 = 2 × 5 = 10

• T. True
• F. False

Answer: True

This correctly applies the product rule: √(ab) = √a · √b. Since 4 and 25 are both perfect squares and are being multiplied (not added), the rule applies. You can verify: 4 × 25 = 100, and √100 = 10. ✓

Question 5

Explain why finding the largest perfect square factor of a radicand is more efficient than finding any perfect square factor, using √72 as an example.

Model answer: Finding the largest perfect square factor simplifies a radical in one step. For √72, the largest perfect square factor is 36, giving √72 = √(36 × 2) = 6√2 immediately. Using a smaller factor like 4 gives √72 = 2√18, but √18 still contains a perfect square (9), requiring a second application of the product rule to reach 6√2. The largest factor eliminates all perfect squares in one step.

The product rule can always be applied repeatedly — so any perfect square factor will eventually lead to the correct simplified form. But the largest perfect square factor gets there directly. This matters in multi-step problems where an unsimplified radical leads to errors in later calculations. Building fluency with perfect squares (4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144) makes spotting the largest factor fast.