Questions: Operations with Radicals

Question 1

A student wants to simplify √12 + √75. What must they do first, and what is the final simplified result?

• A. Add the radicands: √12 + √75 = √87
• B. Simplify each radical first: √12 = 2√3 and √75 = 5√3, then combine like radicals to get 7√3
• C. Multiply using the product rule: √12 × √75 = √900 = 30
• D. Recognize that 12 and 75 are different radicands, so the expression cannot be simplified

Answer: (B)

The critical first step is simplification: √12 = √(4×3) = 2√3, and √75 = √(25×3) = 5√3. Once simplified, both radicals share the radicand 3 — they are like radicals — and combine exactly like like terms: 2√3 + 5√3 = 7√3. Option A is the most dangerous misconception: adding radicands (√12 + √75 ≠ √87) violates the rule that you can only combine radicals with matching radicands. Option D is a common mistake when students skip the simplification step.

Question 2

Which of the following is correct?

• A. √3 + √5 = √8, because you add the radicands just as you add the numbers under the radical
• B. √3 × √5 = √15, because the product rule allows multiplication of unlike radicals
• C. √3 + √5 = √15, because you multiply the radicands when the radicals are unlike
• D. √3 × √5 cannot be simplified because the radicands are different

Answer: (B)

The product rule states √a × √b = √(ab) for a,b ≥ 0 — unlike radicals CAN be multiplied. √3 × √5 = √15 (which cannot be further simplified). In contrast, √3 + √5 cannot be combined because addition of radicals requires identical radicands. Option A (√3 + √5 = √8) is the most common misconception — adding radicands as if they were ordinary numbers. The operations of multiplication and addition follow completely different rules for radicals: unlike radicals can be multiplied but not added.

Question 3

3√5 + 7√5 = 10√5, for the same reason that 3x + 7x = 10x — in both cases, the coefficients count how many of the same 'unit' you have.

• T. True
• F. False

Answer: True

This analogy is the key conceptual insight: the radical √5 acts exactly like a variable. The coefficients (3 and 7) count how many copies of that unit you have, and like any like-terms combination, you add the coefficients while keeping the shared unit unchanged. 3√5 + 7√5 = (3 + 7)√5 = 10√5, just as 3x + 7x = 10x. The rule only applies when the radicands are identical — you cannot combine 3√5 and 7√3 for the same reason you cannot combine 3x and 7y.

Question 4

√2 + √8 can seldom be simplified because 2 and 8 are different radicands.

• T. True
• F. False

Answer: False

This is incorrect — simplification reveals hidden like radicals. √8 = √(4×2) = 2√2. After simplification, √2 + √8 = √2 + 2√2 = 3√2. The lesson: always simplify each radical fully before concluding that radicals are unlike. Two radicals may appear to have different radicands but become like radicals after simplification. The procedure is: simplify first, then check for matching radicands, then combine.

Question 5

Explain why √12 + √27 can be simplified even though 12 and 27 are different numbers. What makes two radicals 'like radicals,' and why does simplification matter?

Model answer: √12 = 2√3 (because 12 = 4 × 3, and √4 = 2) and √27 = 3√3 (because 27 = 9 × 3, and √9 = 3). After simplification, both radicals share the radicand 3, making them like radicals. Like radicals are those with identical radicands after full simplification — they function as the same unit. Since 2√3 + 3√3 = (2+3)√3 = 5√3, the sum simplifies. Without simplifying first, the different surface radicands (12 and 27) make the radicals appear unlike, leading to the wrong conclusion that they cannot be combined.

Simplification is the unlocking step. Different radicands do not necessarily mean unlike radicals — they may share a common simplified form. The reliable procedure is always: fully simplify each radical by extracting the largest perfect-square factor, then check whether the simplified radicands match. Only after this step can you correctly determine whether addition or subtraction is possible.