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Let's solve the problem step by step.

### Part (a): 
\[ \left( 5.1 \times 10^{-3} \right) + \left( 3.82 \times 10^{-1} \right) \]

To add these numbers, both should have the same exponent. We'll convert \( 3.82 \times 10^{-1} \) to \( 0.382 \times 10^0 \), which is:

\[ 5.1 \times 10^{-3} = 0.0051 \]
\[ 3.82 \times 10^{-1} = 0.382 \]

Now, perform the addition:

\[
0.0051 + 0.382 = 0.3871
\]

Convert the result back to scientific notation:

\[
0.3871 = 3.871 \times 10^{-1}
\]

Since the smallest number of significant figures is 2 (from \( 5.1 \times 10^{-3} \)), we round the result to 2 significant figures:

\[
\boxed{3.9 \times 10^{-1}}
\]

---

### Part (b): 
\[ \log \left( \frac{5.57 \times 10^2}{2.13 \times 10^4} \right) \]

First, simplify the expression inside the logarithm:

\[
\frac{5.57 \times 10^2}{2.13 \times 10^4} = \frac{5.57}{2.13} \times 10^{2 - 4} = \frac{5.57}{2.13} \times 10^{-2}
\]

Perform the division:

\[
\frac{5.57}{2.13} \approx 2.615
\]

So, we have:

\[
2.615 \times 10^{-2}
\]

Now apply the logarithmic property:

\[
\log (2.615 \times 10^{-2}) = \log (2.615) + \log (10^{-2}) = \log (2.615) - 2
\]

Calculate the logarithm of 2.615:

\[
\log (2.615) \approx 0.417
\]

Thus, the result is:

\[
0.417 - 2 = -1.583
\]

Rounding to three significant figures (matching the significant figures of 5.57 and 2.13):

\[
\boxed{-1.58}
\]

---

### Summary:
- **Part (a):** \( 3.9 \times 10^{-1} \)
- **Part (b):** \( -1.58 \)

Let me know if you'd like further clarification or more details!

---

#### Related Questions:
1. What are the general rules for adding numbers in scientific notation?
2. How does one handle significant figures in logarithmic calculations?
3. How can you quickly estimate the value of logarithms for non-integer numbers?
4. What is the difference between natural log and common log?
5. Why is it important to express scientific answers with the correct number of significant figures?

**Tip:** Always match the number of significant figures in your final result to the value with the fewest significant figures in the initial data!

Perform the following calculations showing all steps. Express the solution in scientific notation with the appropriate number of significant figures: (a) (5.1 × 10^-3) + (3.82 × 10^-1) (b) log((5.57 × 10^2) / (2.13 × 10^4))

Learn how to perform addition in scientific notation and logarithmic calculations, ensuring the correct number of significant figures. This detailed example covers (a) addition of (5.1 × 10^-3) + (3.82 × 10^-1) and (b) log((5.57 × 10^2) / (2.13 × 10^4)) with all steps shown.

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