To find the distance from the point [tex]\((8, 7, -5)\)[/tex] to the origin [tex]\((0, 0, 0)\)[/tex] in three-dimensional space, we use the Euclidean distance formula. The formula for the distance [tex]\(d\)[/tex] between two points [tex]\((x_1, y_1, z_1)\)[/tex] and [tex]\((x_2, y_2, z_2)\)[/tex] in three-dimensional space is:
[tex]\[ d = \sqrt{{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}} \][/tex]
Since the origin is [tex]\((0, 0, 0)\)[/tex], the formula simplifies to:
[tex]\[ d = \sqrt{{x^2 + y^2 + z^2}} \][/tex]
Plugging in the coordinates of the point [tex]\((8, 7, -5)\)[/tex]:
[tex]\[ d = \sqrt{8^2 + 7^2 + (-5)^2} \][/tex]
Now, calculate each term inside the square root:
[tex]\[ 8^2 = 64 \][/tex]
[tex]\[ 7^2 = 49 \][/tex]
[tex]\[ (-5)^2 = 25 \][/tex]
Next, sum these values:
[tex]\[ 64 + 49 + 25 = 138 \][/tex]
Now take the square root of the sum:
[tex]\[ d = \sqrt{138} \][/tex]
The exact value of [tex]\(\sqrt{138}\)[/tex] is approximately [tex]\(11.74734012447073\)[/tex].
Rounding this to the nearest hundredth, we get:
[tex]\[ d \approx 11.75 \][/tex]
Therefore, the distance from the point [tex]\((8, 7, -5)\)[/tex] to the origin, rounded to the nearest hundredth, is:
[tex]\[ \boxed{11.75} \][/tex]
