Question

find the distance between the pair of points $(4\sqrt{2},\sqrt{5})$ and $(-\sqrt{2},5\sqrt{5})$. if necessary, express the answer in simplified radical form and then round to two decimal places.
the distance between the given points is $\square$ units.
(simplify your answer, using radicals as needed.)
the distance between the given points is approximately $\square$ units.
(type an integer or decimal rounded to two decimal places as needed.)

Explanation:

Step1: Use the distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
Here, \(x_1 = 4\sqrt{2}\), \(y_1=\sqrt{5}\), \(x_2=-\sqrt{2}\), \(y_2 = 5\sqrt{5}\).
Substitute these values into the formula:

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Step2: Simplify the expressions inside the square - root

First, simplify \(-\sqrt{2}-4\sqrt{2}=-5\sqrt{2}\) and \(5\sqrt{5}-\sqrt{5} = 4\sqrt{5}\).
Then \(d=\sqrt{(-5\sqrt{2})^2+(4\sqrt{5})^2}\).
Using the property \((ab)^2=a^{2}b^{2}\), we have \((-5\sqrt{2})^2=(-5)^2\times(\sqrt{2})^2 = 25\times2=50\) and \((4\sqrt{5})^2=4^{2}\times(\sqrt{5})^2=16\times5 = 80\).
So \(d=\sqrt{50 + 80}=\sqrt{130}\)

Step3: Approximate the value

\(\sqrt{130}\approx11.40\) (rounded to two decimal places)

Answer:

The distance between the given points is \(\sqrt{130}\) units. The distance between the given points is approximately \(11.40\) units.