Find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places. (7/3, 1/5) and (1/3, 6/5)

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Identify the coordinates of the two points: \(\left(\frac{7}{3}, \frac{1}{5}\right)\) and \(\left(\frac{1}{3}, \frac{6}{5}\right)\).

Recall the distance formula between two points \(\left(x_1, y_1\right)\) and \(\left(x_2, y_2\right)\): \[d = \sqrt{\left(x_2 - x_1\right)^2 + \left(y_2 - y_1\right)^2}\]

Substitute the given coordinates into the formula: \[d = \sqrt{\left(\frac{1}{3} - \frac{7}{3}\right)^2 + \left(\frac{6}{5} - \frac{1}{5}\right)^2}\]

Simplify the differences inside the parentheses: \[\left(\frac{1}{3} - \frac{7}{3}\right) = \frac{1 - 7}{3} = \frac{-6}{3}\] and \[\left(\frac{6}{5} - \frac{1}{5}\right) = \frac{6 - 1}{5} = \frac{5}{5}\]

Square each simplified difference, add them, and then take the square root to find the distance. Express the result in simplified radical form before rounding to two decimal places.

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주요 개념

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Distance Formula

The distance formula calculates the length between two points in a coordinate plane. It is derived from the Pythagorean theorem and given by d = √((x2 - x1)² + (y2 - y1)²). This formula helps find the straight-line distance between points (x1, y1) and (x2, y2).

Simplifying Radical Expressions

Simplifying radicals involves expressing a square root in its simplest form by factoring out perfect squares. This makes the answer cleaner and easier to interpret before rounding. For example, √50 can be simplified to 5√2 by factoring 50 into 25 × 2.

Rounding Decimal Numbers

Rounding is the process of approximating a number to a specified decimal place for simplicity. In this problem, answers are rounded to two decimal places, meaning the number is truncated or increased based on the third decimal digit. This makes results easier to read and use.