Question

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

(-4, 4) and (2, -4)

Explanation:

Step1: Recall distance formula

The distance \( d \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

Step2: Identify coordinates

Let \((x_1, y_1)=(-4, 4)\) and \((x_2, y_2)=(2, -4)\).

Step3: Substitute into formula

Calculate \( x_2 - x_1 = 2 - (-4)=6 \) and \( y_2 - y_1=-4 - 4=-8 \).
Then \( d=\sqrt{(6)^2+(-8)^2}=\sqrt{36 + 64} \).

Step4: Simplify the radical

\(\sqrt{36 + 64}=\sqrt{100}=10\)? Wait, no, wait: \( 36+64 = 100 \)? Wait, no, \( 6^2=36 \), \( (-8)^2 = 64 \), \( 36 + 64=100 \), so \( \sqrt{100}=10 \)? Wait, but let's check again. Wait, \( 2 - (-4)=6 \), correct. \( -4 - 4=-8 \), correct. Then \( 6^2 = 36 \), \( (-8)^2=64 \), sum is \( 100 \), square root of \( 100 \) is \( 10 \). Wait, but maybe I made a mistake? Wait, no, the distance formula is correct. So the distance is \( 10 \)? Wait, but let's re - calculate:

Wait, \( (x_2 - x_1)=2-(-4)=2 + 4 = 6 \), \( (y_2 - y_1)=-4 - 4=-8 \). Then \( (x_2 - x_1)^2=6^2 = 36 \), \( (y_2 - y_1)^2=(-8)^2 = 64 \). Sum is \( 36+64 = 100 \). Square root of \( 100 \) is \( 10 \). So the distance is \( 10 \).

Wait, but the problem says "simplest radical form". But \( \sqrt{100}=10 \), which is an integer, so that's the simplest form.

Answer:

\( 10 \)