Question: What is the base-ten equivalent of the base-seven number $ 321_7 $?

Question: What is the base-ten equivalent of the base-seven number $ 321_7 $?

["What is the Base-Ten Equivalent of the Base-Seven Number $ 321_7 $?", "When working with numbers in different bases, converting from a radical system like base-seven to base-ten is a fundamental skill in mathematics, computer science, and digital systems. One commonly encountered question is: What is the base-ten equivalent of the base-seven number $ 321_7 $?", "### Understanding Base-Seven Representation", "In base-seven, each digit represents a power of seven, increasing from right to left starting with $ 7^0 $. The number $ 321_7 $ consists of the digits:", "- $ 3 $ in the $ 7^2 $ (hundreds) place\n- $ 2 $ in the $ 7^1 $ (sevens) place\n- $ 1 $ in the $ 7^0 $ (ones) place", "To convert this to base-ten, we use the positional notation formula:", "$$\n321_7 = 3 \ imes 7^2 + 2 \ imes 7^1 + 1 \ imes 7^0\n$$", "### Performing the Calculation", "We now evaluate each term step-by-step:", "1. $ 7^2 = 49 $, so $ 3 \ imes 49 = 147 $\n2. $ 7^1 = 7 $, so $ 2 \ imes 7 = 14 $\n3. $ 7^0 = 1 $, so $ 1 \ imes 1 = 1 $", "Adding these together:", "$$\n147 + 14 + 1 = 162\n$$", "### Final Result", "Thus, the base-ten equivalent of $ 321_7 $ is:", "$$\n\boxed{162}\n$$", "### Why This Conversion Matters", "Understanding base conversions is essential in fields such as computer science (where base-2, base-8, and base-16 dominate), digital electronics, cryptography, and algorithm design. The ability to fluently convert from base-seven (or any base) to base-ten strengthens logical reasoning and computational thinking.", "Whether you're solving math problems, working with non-decimal systems, or studying numeral representations, mastering base conversions—like converting $ 321_7 $ to base-ten—builds a solid foundation for advanced topics.", "---", "Keywords: base-seven to base-ten, convert $ 321_7 $, base conversions, math tutorial, numeral systems, digits in base-seven, positional notation\nMeta Description: Learn how to convert $ 321_7 $ to base-ten with step-by-step calculation and explanation — essential for math students and computing fundamentals."]

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\frac{x + y}{x - y} + \frac{y + z}{y - z} + \frac{z + x}{z - x}given that \( x > y > z \).We are to minimize the expressionS = \frac{x + y}{x - y} + \frac{y + z}{y - z} + \frac{z + x}{z - x}under the condition \( x > y > z > 0 \). Note that all denominators are positive since \( x > y > z \), so each fraction is positive and unbounded as any variable approaches another.We analyze the structure of each term. Consider the identity:\frac{x + y}{x - y} = \frac{(x - y) + 2y}{x - y} = 1 + \frac{2y}{x - y}\frac{y + z}{y - z} = 1 + \frac{2z}{y - z}, \quad \frac{z + x}{z - x} = \frac{z + x}{-(x - z)} = -\frac{z + x}{x - z} = -\left( \frac{(x - z) + 2z}{x - z} \right) = -1 - \frac{2z}{x - z}Adding the three:S = \left(1 + \frac{2y}{x - y}\right) + \left(1 + \frac{2z}{y - z}\right) + \left(-1 - \frac{2z}{x - z}\right)