### Instruction: Find the distance between the points $(2,2)$ and $(-1,-1)$.### Response: To find the distance between two points, I can use the distance formula, which is derived from the Pythagorean theorem.
The distance formula is $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$, where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two points.
In this case, I can label $(2,2)$ as $(x_1, y_1)$ and $(-1,-1)$ as $(x_2, y_2)$.
Then I can plug these values into the formula and simplify.
$d = \sqrt{(-1 - 2)^2 + (-1 - 2)^2}$
$d = \sqrt{(-3)^2 + (-3)^2}$
$d = \sqrt{9 + 9}$
$d = \sqrt{18}$
I can simplify this radical by factoring out the largest perfect square that divides 18, which is 9.
Then I can use the property that $\sqrt{ab} = \sqrt{a}\sqrt{b}$.
$d = \sqrt{9 \cdot 2}$
$d = \sqrt{9}\sqrt{2}$
$d = 3\sqrt{2}$
This is the exact value of the distance.
If I want to approximate it as a decimal, I can use a calculator or estimate that $\sqrt{2} \approx 1.4$.
$d \approx 3 \cdot 1.4$