In Exercises 1–18, find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places. (0, −√3) and (√5, 0)

Verified step by step guidance

1. Identify the coordinates of the two points: \((x_1, y_1) = (0, -\sqrt{3})\) and \((x_2, y_2) = (\sqrt{5}, 0)\).
2. Use the distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).
3. Substitute the coordinates into the formula: \(d = \sqrt{(\sqrt{5} - 0)^2 + (0 - (-\sqrt{3}))^2}\).
4. Simplify the expression inside the square root: \(d = \sqrt{(\sqrt{5})^2 + (\sqrt{3})^2}\).
5. Calculate the squares: \(d = \sqrt{5 + 3}\).

Key Concepts

Distance Formula

The distance between two points in a Cartesian plane can be calculated using the Distance Formula, which is derived from the Pythagorean theorem. The formula is given by d = √((x2 - x1)² + (y2 - y1)²), where (x1, y1) and (x2, y2) are the coordinates of the two points. This formula allows us to find the straight-line distance between any two points.

Simplified Radical Form

Simplified radical form refers to expressing a square root in its simplest terms, where the radicand (the number under the square root) has no perfect square factors other than 1. For example, √12 can be simplified to 2√3. This form is often required in mathematical answers to ensure clarity and conciseness.

Rounding Numbers

Rounding numbers involves adjusting a number to a nearby value that is simpler or more convenient, typically to a specified number of decimal places. In this context, rounding to two decimal places means keeping only two digits after the decimal point, which helps in presenting answers in a more manageable format while maintaining a reasonable level of precision.