To express $\tan M$ as a fraction in simplest terms, we need to interpret the given values as corresponding to a right triangle.

Let's assume $K$, $L$, and $M$ are the angles, and the sides $32$, $24$, and $40$ represent the triangle's side lengths. Since $M$ is the angle in question, the tangent of $M$ is given by:

$\tan M = \frac{\text{opposite}}{\text{adjacent}}$

From the given values, it seems we have a triangle with sides $32$, $24$, and $40$. Since $40$ is the largest side, it is likely the hypotenuse. We now need to determine which of the remaining sides ($32$ or $24$) is opposite and which is adjacent to angle $M$.

Assuming side $32$ is opposite and side $24$ is adjacent to angle $M$:

$\tan M = \frac{32}{24}$

Now, simplify the fraction:

$\tan M = \frac{32}{24} = \frac{4}{3}$

Thus, $\tan M = \frac{4}{3}$ in simplest terms.

Would you like more details or have any questions?

Here are 5 related questions:

1. What is $\sin M$ using the same triangle?
2. How do you calculate $\cos M$?
3. How do the angles $K$ and $L$ relate in this triangle?
4. How can we verify if this is a right triangle using the Pythagorean theorem?
5. How do you calculate $\tan M$ for angles in non-right triangles?

Tip: Always check if a triangle is right by verifying that the sum of the squares of the two smaller sides equals the square of the hypotenuse!