Find the base five numeral expanded to give (3 x 5 x 5) + (4 x 1) and workout 212three - 121three.

You're on a roll — Here are the solutions to the problems: Problem 1: Find the base five numeral expanded to give (3 × 5 × 5) + (4 × 1) Step 1: Rewrite the expanded form using powers of 5. The given expression is (3 × 5 × 5) + (4 × 1). This can be written as: (3 × 5^2) + (4 × 5^0) Step 2: Identify the coefficients for each power of 5. In base five, a number d_n d_n-1 d_1 d_0 is represented as d_n × 5^n + d_n-1 × 5^n-1 + + d_1 × 5^1 + d_0 × 5^0. Comparing this to our expression: The coefficient for 5^2 is 3. There is no 5^1 term, so its coefficient is 0. The coefficient for 5^0 (which is 1) is 4. Step 3: Construct the base five numeral. The digits are 3 (for 5^2), 0 (for 5^1), and 4 (for 5^0). Therefore, the base five numeral is 304_5. The base five numeral is 304_5. Problem 2: Workout: 212_three - 121_three Step 1: Set up the subtraction in base three. We subtract column by column from right to left. r 212_three \\ -121_three \\ Step 2: Subtract the rightmost digits (units place). 2_three - 1_three = 1_three. r 212_three \\ -121_three \\ 1_three Step 3: Subtract the middle digits (3s place). We need to calculate 1_three - 2_three. Since 1 < 2, we must borrow from the next place value (the 3^2 place). Borrow 1 from the 2_three in the 3^2 place. This borrowed 1 is equivalent to 3_three in the 3^1 place. So, the 1_three in the 3^1 place becomes 1_three + 3_three = 4_three. Now, subtract: 4_three - 2_three = 2_three. The 2_three in the 3^2 place becomes 1_three after borrowing. r 2 1 2_three \\ 1 (1+3) \\ -1 2 1_three \\ 2 1_three Step 4: Subtract the leftmost digits (3^2 place). We now have 1_three - 1_three = 0_three. r 1 4 2_three \\ -1 2 1_three \\ 0 2 1_three Step 5: Combine the results. The result is 021_three, which is written as 21_three. The result of the workout is 21_three. Got more? Send 'em!