What is the value of √75 simplified?

The simplified radical form of √75 equals 5√3 when the radicand 75 is factored into 25 × 3, the perfect square factor 25 is extracted as its square root 5 outside the radical, and the remaining non-square factor 3 remains under the radical symbol, producing the expression 5√3 that cannot be simplified further as 3 contains no perfect square factors beyond 1.

A. 5√3

This expression correctly represents the simplified radical through systematic factorization and extraction verified through multiple approaches. Prime factorization pathway: 75 = 3 × 5²; extract square factor 5² as 5 outside radical; remaining factor 3 stays inside → 5√3. Perfect square identification: largest perfect square dividing 75 is 25; 75 ÷ 25 = 3; √75 = √(25×3) = √25 × √3 = 5√3. Verification by squaring: (5√3)² = 25 × 3 = 75, confirming equivalence. Decimal approximation: √75 ≈ 8.660; 5√3 ≈ 5 × 1.732 = 8.660, confirming numerical equality. This solution demonstrates comprehensive mastery of radical simplification including perfect square factor identification, extraction protocols preserving multiplicative relationships (√ab = √a√b for a,b ≥ 0), recognition of termination conditions (no perfect square factors remain in radicand), and verification through squaring or decimal approximation integrated competencies essential for algebraic manipulation, trigonometric simplification, calculus integration techniques, and physics problem-solving where simplified radical forms enable pattern recognition, term combination, and exact value preservation avoiding decimal approximation errors in multi-step calculations.

B. 3√5

This expression equals √45 rather than √75, as (3√5)² = 9 × 5 = 45. The error likely originates from factoring 75 as 9 × 8.333... then incorrectly identifying 9 as the largest perfect square factor while miscalculating the cofactor as 5 instead of 75 ÷ 9 = 8.333... Students might execute 75 ÷ 3 = 25 then misinterpret the factorization as 3 × 25 and extract √9 = 3 instead of √25 = 5 confusing which factor is the perfect square. Another plausible pathway involves correctly identifying factors 3 and 25 but then extracting the square root of the non-square factor (3) instead of the square factor (25), yielding √3 outside and 5 inside as 5√3 but transposing to 3√5 through digit substitution. The value 3√5 ≈ 6.708 differs substantially from √75 ≈ 8.660 revealing how factor misidentification produces solutions with incorrect magnitude despite superficial structural similarity to correct form. This distractor effectively identifies learners with fragmented factorization knowledge who recognize simplification requires factoring but lack systematic perfect square identification protocols a deficiency requiring explicit largest-perfect-square-first approach with factor pair enumeration (1×75, 3×25, 5×15) to identify 25 as the maximal square factor before extraction.

C. 15√5

This expression equals √1125 rather than √75, as (15√5)² = 225 × 5 = 1125 fifteen times larger than 75. The error likely originates from multiplying factors instead of extracting square roots: recognizing 75 = 25 × 3 but then executing 25 × 3 = 75 and placing 75 outside the radical as 75√1, then simplifying 75 to 15×5 and extracting 5 to yield 15√5 through unstructured manipulation. Another plausible pathway involves solving √75 = x√5 → x = √15 ≈ 3.873 then rounding to 4 and multiplying by incorrect coefficient to reach 15. Students might confuse simplification with rationalization procedures or misapply exponent rules (√75 = 75^½ = (15×5)^½ = 15^½×5^½) then incorrectly converting 15^½ to 15. The dramatic magnitude disparity (15√5 ≈ 33.54 versus √75 ≈ 8.66) reveals fundamental misunderstanding of radical extraction mechanics where learners treat coefficients as multiplicative accumulations rather than square root extractions a critical misconception requiring explicit demonstration that √(a²b) = a√b (not a²√b) through concrete examples showing area models where side length extraction corresponds to square root operation, not area preservation.

D. 25√3

This expression equals √1875 rather than √75, as (25√3)² = 625 × 3 = 1875 twenty-five times larger than 75. The error likely originates from extracting the perfect square factor 25 but forgetting to take its square root, placing 25 outside the radical instead of 5. Students correctly factor 75 = 25 × 3 and recognize 25 as perfect square but then execute √(25×3) = 25√3 instead of √25 × √3 = 5√3 omitting the critical square root operation on the extracted factor. This "forgotten square root" error represents one of the most pervasive mistakes in radical simplification, particularly when large perfect squares appear creating cognitive load that triggers procedural omission. Another plausible pathway involves solving √75 = x√3 → x² = 25 → x = 5 but then misreading x = 5 as x² = 25 and placing 25 outside the radical. The consistent 5× magnitude error (25 versus 5) reveals systematic procedural deficiency rather than random miscalculation diagnostically valuable for identifying learners who understand factorization requirements but lack extraction mechanics precision, a transitional error requiring explicit emphasis on the dual nature of extraction: (1) identify perfect square factor, (2) take its square root before placing outside radical with visual reinforcement using area models showing 25-unit squares having 5-unit sides to concretize the square root relationship.

Conclusion

The simplified form 5√3 emerges through rigorous factorization identifying 25 as the largest perfect square factor of 75, followed by proper extraction taking the square root of 25 to yield 5 outside the radical while preserving the non-square factor 3 inside. This problem reinforces critical radical manipulation competencies essential across mathematical domains: executing prime factorization to identify perfect square components, applying the product rule for radicals (√ab = √a√b) with non-negative constraints, recognizing termination conditions when radicands contain no perfect square factors beyond 1, and verifying simplifications through squaring to confirm equivalence with original radicand. Mastery of these integrated skills proves indispensable for trigonometry (special angle simplification), calculus (integration requiring radical manipulation), physics (vector magnitude calculations), and engineering (structural load computations) where simplified radical forms enable exact value preservation, term combination in algebraic expressions, and pattern recognition in mathematical relationships capabilities compromised by decimal approximations that introduce rounding errors in multi-step problem-solving sequences. The distractors strategically target pervasive error patterns including factor misidentification, coefficient multiplication instead of extraction, and omitted square root operations during extraction highlighting the necessity of explicit two-step extraction protocols ("factor, then root") with area model visualizations concretizing the geometric meaning of square roots as side lengths of square areas to build robust conceptual foundations supporting accurate radical manipulation across increasingly complex algebraic contexts.