Number System

3 and 4

Step 1: Find perfect squares near 10 to estimate √10. Step 2: We know that 3² = 9 and 4² = 16. Step 3: Since 9 < 10 < 16, we have √9 < √10 < √16. Step 4: Therefore, 3 < √10 < 4, so √10 lies between consecutive integers 3 and 4.

-2/3

Step 1: To express a rational number in standard form, ensure the denominator is positive and the numerator and denominator are coprime. Step 2: Find GCD of 24 and 36: factors of 24 = 1,2,3,4,6,8,12,24 and factors of 36 = 1,2,3,4,6,9,12,18,36. GCD = 12. Step 3: Divide both numerator and denominator by 12: -24/36 = (-24÷12)/(36÷12) = -2/3. Step 4: The denominator is positive, and GCD of 2 and 3 is 1, so -2/3 is in standard form.

35/99

Step 1: Let x = 0.353535... = 0.35̄. Step 2: Multiply both sides by 100 (since the repeating block has 2 digits): 100x = 35.353535... Step 3: Subtract the original equation: 100x - x = 35.353535... - 0.353535..., so 99x = 35. Step 4: Therefore, x = 35/99. We can verify this is in lowest terms since GCD(35,99) = 1.

Addition

Step 1: Check each operation with rational numbers a/b and c/d where b,d ≠ 0. Step 2: Addition: (a/b) + (c/d) = (ad + bc)/(bd), which is rational since ad+bc and bd are integers with bd ≠ 0. Step 3: Square root: √(a/b) may be irrational (e.g., √(1/2) = √2/2). Step 4: Taking reciprocal of zero is undefined, and raising to certain powers can yield irrationals. Therefore, only addition always gives a rational result.