Real Numbers

2√2

Step 1: To simplify √8, find the largest perfect square factor of 8. Step 2: 8 = 4 × 2, where 4 is a perfect square. Step 3: √8 = √(4 × 2) = √4 × √2 = 2 × √2 = 2√2. Step 4: Check: (2√2)² = 4 × 2 = 8. ✓ Correct! Final Step: √8 in simplest form is 2√2. Note: 2√4 = 2×2 = 4, which is wrong as √4 can be simplified.

21√3

Step 1: These are like surds (same radicand √3), so they can be added/subtracted directly. Step 2: Treat √3 as a common factor: (5 + 18 - 2)√3. Step 3: Calculate the coefficients: 5 + 18 = 23, then 23 - 2 = 21. Step 4: So the result is 21√3. Final Step: Just like 5x + 18x - 2x = 21x, we get 5√3 + 18√3 - 2√3 = 21√3.

0.101001000100001...

Step 1: A rational number has either a terminating or non-terminating recurring decimal expansion. Step 2: 0.25 is terminating → rational. 0.3333... is non-terminating recurring → rational. Step 3: 0.76̄ = 0.7666... is non-terminating recurring → rational. Step 4: 0.101001000100001... has a pattern but the digits do NOT recur in a fixed block — it is non-terminating and non-recurring. Final Step: A non-terminating, non-recurring decimal is always irrational.

4.56 × 10⁻⁴

Step 1: In scientific notation, write the number as a × 10ⁿ where 1 ≤ a < 10. Step 2: Move the decimal point to get a number between 1 and 10: 0.000456 → 4.56. Step 3: Count how many places we moved the decimal to the RIGHT: 4 places (0.000456 → 4.56). Step 4: Since we moved right, the exponent is negative: n = -4. Final Step: 0.000456 = 4.56 × 10⁻⁴. Note: 45.6 × 10⁻⁵ is not valid as 45.6 > 10.