Find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places. (0, -√2) and (√7,0)

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Identify the coordinates of the two points: Point 1 is \(\left(0, -\sqrt{2}\right)\) and Point 2 is \(\left(\sqrt{7}, 0\right)\).

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 distance formula: \[d = \sqrt{\left(\sqrt{7} - 0\right)^2 + \left(0 - (-\sqrt{2})\right)^2}\]

Simplify inside the square root by squaring the differences: \[d = \sqrt{\left(\sqrt{7}\right)^2 + \left(\sqrt{2}\right)^2}\]

Calculate the squares and combine under the square root to express the distance in simplified radical form before rounding.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Distance Formula

The distance formula calculates the length between two points in the 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 square roots in their simplest form by factoring out perfect squares. This process makes the answer more exact and easier to interpret before rounding. For example, √8 can be simplified to 2√2 by factoring 8 as 4×2.

Rounding Decimal Numbers

Rounding is the process of approximating a number to a specified number of decimal places for simplicity and clarity. In this problem, after simplifying the radical, the distance should be rounded to two decimal places, ensuring the answer is both precise and easy to use.

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