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: Point 1 is \(\left(3\sqrt{3}, \sqrt{5}\right)\) and Point 2 is \(\left(-\sqrt{3}, 4\sqrt{5}\right)\).
2. Recall the distance formula between two points \(\left(x_1, y_1\right)\) and \(\left(x_2, y_2\right)\): \(d = \sqrt{\left(x_2 - x_1\right)^2 + \left(y_2 - y_1\right)^2}\).
3. Substitute the given coordinates into the distance formula: \(d = \sqrt{\left(-\sqrt{3} - 3\sqrt{3}\right)^2 + \left(4\sqrt{5} - \sqrt{5}\right)^2}\).
4. Simplify inside the parentheses: \(x\)-difference: \(-\sqrt{3} - 3\sqrt{3} = -4\sqrt{3}\) \(y\)-difference: \$4\sqrt{5} - \sqrt{5} = 3\sqrt{5}$.
5. Square each difference and add them under the square root: \(d = \sqrt{\left(-4\sqrt{3}\right)^2 + \left(3\sqrt{5}\right)^2} = \sqrt{16 \times 3 + 9 \times 5}\).

Key Concepts

Distance Formula

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

Simplifying Radicals

Simplifying radicals involves expressing square roots in their simplest form by factoring out perfect squares. This process makes the expression easier to interpret and compare. For example, √12 can be simplified to 2√3 by factoring 12 into 4 × 3.

Rounding Decimal Values

Rounding is the process of approximating a number to a specified degree of accuracy, such as two decimal places. After calculating an exact value, rounding makes the result more practical for interpretation and communication, especially when dealing with irrational numbers.