.3 Determine, without the use of a calculator, between which two .4 Prove that 0.43 is rational. .5 Use this format \( : \frac{a^{2}-1}{a-1} \) to simplify the expression, WITHOUT using calculator: \( \frac{2024}{44} \)

Below are solutions to each part: ───────────────────────────── Part .3 It appears the intended question is to determine (without a calculator) between which two consecutive integers the square root of 2024 lies. Notice that   44² = 1936  and  45² = 2025. Since   1936 < 2024 < 2025, we have   44 < √2024 < 45. Thus, √2024 lies between 44 and 45. ───────────────────────────── Part .4 To prove that 0.43 is rational, recall that a number is rational if it can be written as a fraction of two integers. Since 0.43 is a terminating decimal, we can write it as   0.43 = 43/100. Both 43 and 100 are integers, so 0.43 is a rational number. ───────────────────────────── Part .5 We are asked to simplify   2024/44 using the format   (a² – 1)/(a – 1). Observe that   45² = 2025 → 45² – 1 = 2025 – 1 = 2024, and also notice that   45 – 1 = 44. Thus, we can express   2024/44 = (45² – 1)/(45 – 1). Now, factor the numerator using the difference of squares formula:   a² – 1 = (a – 1)(a + 1). Taking a = 45, we have:   (45² – 1)/(45 – 1) = [(45 – 1)(45 + 1)]/(45 – 1). Canceling the common factor (45 – 1) (which is valid since 45 – 1 ≠ 0), we obtain:   45 + 1 = 46. Thus, 2024/44 simplifies exactly to 46. ───────────────────────────── Summary of Answers: • Part .3: √2024 lies between 44 and 45. • Part .4: 0.43 = 43/100, so it is rational. • Part .5: 2024/44 simplifies to 46.