M varies jointly as x2y and inversely as Z. If M=64 when x=2, y=4, and Z=1, find x when M=144, y=9, and Z=3.

The problem states that M varies jointly as $x^2y$ and inversely as Z. This can be written as:

$$M = k \frac{x^2y}{Z}$$

where $k$ is the constant of proportionality.

We are given $m=64$ when $x=2$, $y=4$, and $Z=1$. Substitute these values into the equation:

$$64 = k \frac{(2)^2(4)}{1}$$ $$64 = k \frac{4 \times 4}{1}$$ $$64 = k \times 16$$

To find $k$, divide both sides by 16:

$$k = \frac{64}{16}$$ $$k = 4$$

Now that we have the value of $k$, the specific relationship between M, x, y, and Z is:

$$M = 4 \frac{x^2y}{Z}$$

We need to find $x$ when $m=144$, $y=9$, and $Z=3$. Substitute these values into the complete variation equation:

$$144 = 4 \frac{x^2(9)}{3}$$

Simplify the right side of the equation:

$$144 = 4 \times (3x^2)$$ $$144 = 12x^2$$

Divide both sides by 12:

$$x^2 = \frac{144}{12}$$ $$x^2 = 12$$

Take the square root of both sides to find $x$:

$$x = \sqrt{12}$$

Simplify the square root:

$$x = \sqrt{4 \times 3}$$ $$x = 2\sqrt{3}$$

The value of $x$ is $\boxed{2\sqrt{3}}$.