Squares and Square Roots

Both radicands are non-negative, so the product rule √a × √b = √(ab) can be used. Therefore √27 × √12 = √(27×12) = √324. Since 324 = 18², its principal square root is 18, so option A is correct. A separate simplification confirms this: √27 = 3√3 and √12 = 2√3; multiplying gives (3√3)(2√3) = 6(√3)² = 6×3 = 18. Option B is twice the correct value. Option C is not the result of the product and is included as a numerical distractor. Option D incorrectly combines the radicands and does not follow the square-root product rule. The exact simplified answer is therefore 18.

The governing concept is extraction of the greatest perfect-square factor from a radicand. Factor 675 as 225 × 3, and recognise that 225 = 15². Therefore, √675 = √(225 × 3) = √225 × √3 = 15√3. Hence option A is the correct simplified surd form. Option C, 3√75, has the same value as √675, but it is not fully simplified because 75 still contains the perfect-square factor 25; simplifying it gives 3 × 5√3 = 15√3. Option B is incorrect because (25√3)² = 1875, not 675. Option D is also too large. Squaring option A gives (15√3)² = 225 × 3 = 675, confirming the answer.

The governing concept is simplifying a radical by taking the largest convenient perfect-square factor outside the square root. Since 980 = 196 × 5 and 196 = 14², we have √980 = √(196 × 5) = √196 × √5 = 14√5. Thus option A is correct. Option B, 7√20, is numerically equal to the original expression, but it is not in simplest surd form because √20 = √(4 × 5) = 2√5, so 7√20 becomes 14√5. Option C is incorrect because its square is 28² × 5 = 3920. Option D has both an incorrect coefficient and radicand. Squaring option A gives 14² × 5 = 196 × 5 = 980, which verifies the result.

The governing concept is simplifying radicals before adding like surds. The first term is already √5. For the second term, √20 = √(4×5) = 2√5. For the third term, √45 = √(9×5) = 3√5. Thus the expression becomes √5 + 2√5 + 3√5 = (1 + 2 + 3)√5 = 6√5. Therefore option A is correct. The coefficients of like surds may be added just like coefficients of algebraic terms, while the common radical √5 remains unchanged. Option B results from an incorrect coefficient sum, and option C uses another incorrect sum. Option D incorrectly combines the radicands as though addition of square roots followed a multiplication rule; in general, √a + √b is not √(ab) or another single radical formed by direct radicand manipulation.