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. (3√3, √5) and (−√3, 4√5)

Verified step by step guidance

1. Identify the coordinates of the two points: \((x_1, y_1) = (3\sqrt{3}, \sqrt{5})\) and \((x_2, y_2) = (-\sqrt{3}, 4\sqrt{5})\).
2. Use the distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).
3. Substitute the coordinates into the formula: \(d = \sqrt{((-\sqrt{3}) - (3\sqrt{3}))^2 + ((4\sqrt{5}) - (\sqrt{5}))^2}\).
4. Simplify the expressions inside the square root: \((-\sqrt{3} - 3\sqrt{3}) = -4\sqrt{3}\) and \((4\sqrt{5} - \sqrt{5}) = 3\sqrt{5}\).
5. Calculate the squares: \((-4\sqrt{3})^2 = 16 \times 3\) and \((3\sqrt{5})^2 = 9 \times 5\), then add them under the square root.

Key Concepts

Distance Formula

The distance formula is a mathematical equation used to determine the distance between two points in a Cartesian coordinate system. It is derived from the Pythagorean theorem and is expressed as d = √((x2 - x1)² + (y2 - y1)²), where (x1, y1) and (x2, y2) are the coordinates of the two points. This formula allows for the calculation of the straight-line distance between any two points in a plane.

Simplified Radical Form

Simplified radical form refers to the expression of a square root in its simplest terms, where no perfect square factors remain under the radical sign. For example, √18 can be simplified to 3√2, as 18 = 9 × 2 and √9 = 3. This form is often preferred in mathematics for clarity and ease of further calculations.

Rounding Numbers

Rounding numbers is the process of adjusting the digits of a number to make it simpler while retaining its approximate value. When rounding to two decimal places, you look at the third decimal place to determine whether to round up or down. This is important in providing a concise answer, especially in contexts where precision is necessary, such as in measurements or calculations.