Question

find the distance between the points (7, 0) and (4, 6). write your answer as a whole number or a fully simplified radical expression. do not round. \sqrt{\box} units

Explanation:

Step1: Recall the distance formula

The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \). Here, \( x_1 = 7,y_1 = 0,x_2 = 4,y_2 = 6 \).

Step2: Substitute the values into the formula

First, calculate \( x_2 - x_1=4 - 7=- 3 \) and \( y_2 - y_1=6 - 0 = 6 \). Then, find the squares: \( (-3)^2 = 9 \) and \( 6^2=36 \). Add these squares: \( 9 + 36=45 \). So the distance is \( \sqrt{45} \), but we can simplify the number inside the square root. Wait, no, the question asks for the number inside the square root box. Wait, the formula gives \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}=\sqrt{(-3)^2 + 6^2}=\sqrt{9 + 36}=\sqrt{45} \), but the box is for the number under the square root. Wait, actually, let's re - check. The distance formula is \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \), so the number inside the square root is \( (x_2 - x_1)^2+(y_2 - y_1)^2 \). So \( (4 - 7)^2+(6 - 0)^2=(-3)^2+6^2 = 9 + 36=45 \). Wait, but maybe I made a mistake? Wait, \( 7-4 = 3 \), \( 0 - 6=-6 \), but squaring removes the sign. So \( (3)^2+(-6)^2=9 + 36 = 45 \). So the number inside the square root is 45. Wait, but let's check again. The two points are \( (7,0) \) and \( (4,6) \). The difference in x - coordinates is \( 4 - 7=-3 \), difference in y - coordinates is \( 6 - 0 = 6 \). Then, \( (-3)^2=9 \), \( 6^2 = 36 \), sum is \( 9+36 = 45 \). So the number inside the square root is 45.

Answer:

45