Surds and Indices

Column A simplifies to sqrt((z²-4)²) = |z²-4|, which is the absolute value of Column B. When z²-4 ≥ 0 (i.e., |z| ≥ 2), both columns are equal. When z²-4 < 0 (i.e., -2 < z < 2), Column A equals 4-z² which is positive, while Column B is negative. Since the relationship changes based on z's value, it cannot be determined universally.

The cube root of 60698457 is 393, since 393 × 393 × 393 = 60698457. This is a numerical computation question requiring you to identify which option cubed equals the given number. Note the grammar issue in the question.

The square root of 289 is 17 because 17 × 17 = 289. Option B is correct. Option A (19) gives 361. Option C (13) gives 169. Option D (28.9) is not an integer and seems to be a decimal approximation error. Memorizing perfect squares up to 20×20 helps with quick mental math.

Every positive real number has exactly two square roots: one positive and one negative. For example, the square roots of 9 are 3 and -3, since both 3^2 = 9 and (-3)^2 = 9. By convention, the radical symbol √ denotes the principal (non-negative) square root, but both roots exist.

If A = √8, then A^2 = (√8)^2 = 8. This is a fundamental property of square roots: squaring a square root returns the original number (for non-negative numbers). The square of √8 cannot involve √2 because it cancels out completely.

√18 = √(9 × 2) = √9 × √2 = 3√2. This simplification uses the property √(ab) = √a × √b. We look for the largest perfect square factor of 18, which is 9, leaving 2 inside the radical. The answer is 3√2, not 2√3, 6√3, or 9√2.

ABC = √6 × √10 × √15 = √(6 × 10 × 15) = √900 = 30. When multiplying square roots, multiply the numbers under the radical signs first, then take the square root of the product.

√12 ÷ √3 = √(12/3) = √4 = 2. When dividing square roots, divide the radicands first, then simplify the resulting square root. The key is recognizing that 12 divided by 3 is 4, which is a perfect square.

1/(4 + √15) rationalizes by multiplying numerator and denominator by the conjugate (4 - √15): [1 × (4 - √15)] / [(4 + √15)(4 - √15)] = (4 - √15) / (16 - 15) = (4 - √15) / 1 = 4 - √15. Rationalizing denominators eliminates radicals from fractions.

a² = (1 + √3)² = 1² + 2(1)(√3) + (√3)² = 1 + 2√3 + 3 = 4 + 2√3. Use the perfect square formula (a + b)² = a² + 2ab + b². Don't forget the middle term 2ab - that's where the 2√3 comes from.

The question gives us (2 + √5)² = 9 + 4√5 and asks for √(9 + 4√5). Since we're taking the square root of a perfect square, we get back the original expression: √(9 + 4√5) = 2 + √5. This is a reverse operation question.

Find the square root of 3249. Testing options: 57 × 57 = 3249. So 57 is the correct square root.

√0.4 ≈ 0.6325. Among the options, 0.636 is closest: 0.636² = 0.4045 ≈ 0.4. Option B (0.61) gives 0.61² = 0.3721, which is less accurate.

64 squared is 4096 (60^2 = 3600, 70^2 = 4900, so it must be between 60 and 70). 32 squared is 1024, and 128 squared is 16384. 54 squared is 2916.

The square root of 2 is a mathematical constant approximately equal to 1.41421. 1.735 is close to the square root of 3, while 1.448 and 1.659 are incorrect approximations.