To convert \(321_9\) to base ten, expand using powers of 9:
["How to Convert (321_9) to Base Ten: Expanding Using Powers of 9", "Converting numbers from one base to base ten can seem challenging at first, but with a clear understanding of place values, the process becomes straightforward. In this article, we’ll walk through the step-by-step conversion of (321_9) (a number in base 9) to base ten (decimal). By expanding the number using powers of 9, we’ll see how each digit contributes to the final decimal value.", "---", "### Understanding Base 9", "Base 9 (nine) uses nine distinct digits: (0, 1, 2, 3, 4, 5, 6, 7, 8). Each digit’s position represents a power of 9, starting from the rightmost digit as (9^0 = 1), then (9^1 = 9), (9^2 = 81), and so on.", "The number (321_9) has three digits:\n- The rightmost digit: (1) (place value (9^0))\n- The middle digit: (2) (place value (9^1))\n- The leftmost digit: (3) (place value (9^2))", "---", "### Expanding Using Powers of 9", "To convert (321_9) to base ten, express each digit multiplied by its corresponding power of 9:", "[\n321_9 = 3 \ imes 9^2 + 2 \ imes 9^1 + 1 \ imes 9^0\n]", "Now calculate the powers of 9:", "- (9^0 = 1)\n- (9^1 = 9)\n- (9^2 = 81)", "Substitute these values:", "[\n321_9 = 3 \ imes 81 + 2 \ imes 9 + 1 \ imes 1\n]", "---", "### Performing the Calculations", "Compute each term:", "- (3 \ imes 81 = 243)\n- (2 \ imes 9 = 18)\n- (1 \ imes 1 = 1)", "Add them together:", "[\n243 + 18 + 1 = 262\n]", "---", "### Final Result", "Thus, the base 9 number (321_9) equals:", "[\n\boxed{262_{10}}\n]", "---", "### Summary", "Converting (321_9) to base ten involves recognizing each digit’s positional weight based on powers of 9. By expanding:", "[\n3 \ imes 9^2 + 2 \ imes 9^1 + 1 \ imes 9^0 = 243 + 18 + 1 = 262\n]", "We confirm that (321_9 = 262_{10}). This method ensures accuracy and helps build a strong foundation in number base conversions."]
