Question

1. find the distance between the two points in simplest radical form.

Explanation:

1. First, assume the two - point formula:

• The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
• From the graph, assume the first point is \((1, - 1)\) and the second point is \((-5,-9)\). Here, \(x_1 = 1,y_1=-1,x_2=-5,y_2 = - 9\).

1. Then, calculate the differences in \(x\) and \(y\) coordinates:

• Calculate \(x_2 - x_1\):
• \(x_2 - x_1=-5 - 1=-6\).
• Calculate \(y_2 - y_1\):
• \(y_2 - y_1=-9-( - 1)=-9 + 1=-8\).

1. Next, substitute into the distance formula:

• Substitute \(x_2 - x_1=-6\) and \(y_2 - y_1=-8\) into \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
• \(d=\sqrt{(-6)^2+(-8)^2}=\sqrt{36 + 64}\).
• Simplify the expression inside the square - root:
• \(36+64 = 100\), so \(d=\sqrt{100}=10\). But if we want to keep it in the process of radical form before the final simplification:
• \(d=\sqrt{(-6)^2+(-8)^2}=\sqrt{36 + 64}=\sqrt{100}=10\). In radical form before full - simplification, we have \(d=\sqrt{36 + 64}=\sqrt{100}= \sqrt{4\times25}=2\times5 = 10\). The simplest radical form considering the steps is \(d=\sqrt{(-6)^2+(-8)^2}=\sqrt{36 + 64}=\sqrt{100}\).

Answer:

\(\sqrt{100}\) (or \(10\) if full - simplification is allowed)