Given $z=7+3i$, find the product $z\cdot \overline{z}$

Step 1 ：Given the complex number $z=7+3i$, we are asked to find the product $z\cdot \overline{z}$.

Step 2 ：The product of a complex number and its conjugate is given by the formula $z\cdot \overline{z}=|z{|}^2$, where $|z|$ is the magnitude of the complex number.

Step 3 ：The magnitude of a complex number $z=a+bi$ is given by $\sqrt{a^2+b^2}$.

Step 4 ：Substituting the given values into the formula, we get $\sqrt{7^2+3^2}$ which simplifies to approximately 7.615773105863909.

Step 5 ：We then square this magnitude to find the product $z\cdot \overline{z}$, which gives us approximately 58.00000000000001.

Step 6 ：This small discrepancy is likely due to the limitations of numerical precision. Therefore, we can conclude that the product $z\cdot \overline{z}$ is 58.

Step 7 ：Final Answer: $\boxed{{\displaystyle 58}}$