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

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

Step 2 ：The product of a complex number and its conjugate is given by the formula \(z \cdot \bar{z} = |z|^2\), 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 \(z \cdot \bar{z}\), 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 \(z \cdot \bar{z}\) is 58.

Step 7 ：Final Answer: \(\boxed{58}\)