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### Question 4: Perform the following calculations, showing all steps. Express the solution in scientific notation with the appropriate number of significant figures.

#### (a) \((5.1 \times 10^{-3}) + (3.82 \times 10^{-1})\)

1. First, convert the numbers to the same power of 10 for easy addition:
   \[
   5.1 \times 10^{-3} = 0.0051
   \quad \text{and} \quad
   3.82 \times 10^{-1} = 0.382
   \]
   
2. Add these values:
   \[
   0.0051 + 0.382 = 0.3871
   \]
   
3. Express the result in scientific notation:
   \[
   0.3871 = 3.871 \times 10^{-1}
   \]
   
4. Adjust for significant figures: Since \(5.1 \times 10^{-3}\) has 2 significant figures and \(3.82 \times 10^{-1}\) has 3 significant figures, the result should have 2 significant figures.
   \[
   \boxed{3.9 \times 10^{-1}}
   \]

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

1. First, compute the ratio inside the logarithm:
   \[
   \frac{5.57 \times 10^2}{2.13 \times 10^4} = \frac{5.57}{2.13} \times 10^{2 - 4} = 2.615 \times 10^{-2}
   \]
   
2. Now, take the logarithm:
   \[
   \log (2.615 \times 10^{-2}) = \log(2.615) + \log(10^{-2})
   \]
   \[
   \log(2.615) \approx 0.4183 \quad \text{and} \quad \log(10^{-2}) = -2
   \]
   \[
   \log(2.615 \times 10^{-2}) = 0.4183 - 2 = -1.5817
   \]
   
3. Round to the appropriate significant figures: Both \(5.57 \times 10^2\) and \(2.13 \times 10^4\) have 3 significant figures, so the final answer should also have 3 significant figures.
   \[
   \boxed{-1.58}
   \]

---

### Question 5: What is the antilogarithm of 1.245? Report with the appropriate number of significant figures.

1. The antilogarithm of 1.245 is calculated as:
   \[
   \text{Antilog}(1.245) = 10^{1.245} \approx 17.63
   \]
   
2. Since 1.245 has 4 significant figures, the result should also have 4 significant figures.
   \[
   \boxed{17.63}
   \]

---

Would you like further details or have any questions?

Here are 5 related questions to extend your understanding:
1. How do you determine the number of significant figures in addition and subtraction?
2. What are the general rules for handling significant figures in multiplication and division?
3. Why does changing the base of the logarithm change the output of a logarithmic expression?
4. What is the difference between the logarithm and antilogarithm in calculations?
5. How does scientific notation simplify calculations involving very large or small numbers?

**Tip:** When working with logarithms and antilogarithms, always make sure to use the correct base (usually 10 for common logs) and round off your answers according to the number of significant figures from the input values.

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))
What is the antilogarithm of 1.245? Report with the appropriate number of significant figures.

Step-by-step solutions to scientific notation addition, logarithmic calculations, and finding the antilogarithm of a number. Learn how to handle powers of 10 and significant figures in mathematical expressions.

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