Express this product in units of km3 in standard scientific notation and with the appropriate number of significant figures:

(3.4 km) × (4.0 m) × (13.24 × 10−3 mm)

To express the product \((3.4 \, \text{km}) \times (4.0 \, \text{m}) \times (13.24 \times 10^{-3} \, \text{mm})\) in cubic kilometers (\(\text{km}^3\)), we need to first convert all units to kilometers, then perform the multiplication, and finally express the result in standard scientific notation with the appropriate number of significant figures.

### Step 1: Convert all units to kilometers
- \(3.4 \, \text{km}\) is already in kilometers.
- \(4.0 \, \text{m}\) needs to be converted to kilometers. 
  \[
  4.0 \, \text{m} = 4.0 \times 10^{-3} \, \text{km}
  \]
- \(13.24 \times 10^{-3} \, \text{mm}\) needs to be converted to kilometers. 
  There are \(1,000,000\) mm in 1 km, so:
  \[
  13.24 \times 10^{-3} \, \text{mm} = 13.24 \times 10^{-9} \, \text{km}
  \]

### Step 2: Perform the multiplication
Now, multiply the values in kilometers:
\[
(3.4 \, \text{km}) \times (4.0 \times 10^{-3} \, \text{km}) \times (13.24 \times 10^{-9} \, \text{km})
\]
Multiplying these together:
\[
3.4 \times 4.0 \times 13.24 = 179.648
\]
And multiplying the powers of 10:
\[
10^0 \times 10^{-3} \times 10^{-9} = 10^{-12}
\]
Thus, the product is:
\[
179.648 \times 10^{-12} \, \text{km}^3
\]

### Step 3: Express in scientific notation with appropriate significant figures
Since the least precise number in the product has 2 significant figures (from \(4.0 \, \text{m}\)), the result must be rounded to 2 significant figures:
\[
179.648 \times 10^{-12} \, \text{km}^3 \approx 1.8 \times 10^{-10} \, \text{km}^3
\]

### Final Answer:
The product is \(1.8 \times 10^{-10} \, \text{km}^3\).

Would you like further clarification on any steps or more details on specific conversions? 

Here are 5 related questions:
1. How do you convert square meters to square kilometers?
2. What are significant figures and why are they important in scientific notation?
3. How would the answer change if one of the lengths was measured in centimeters?
4. Can you multiply quantities with different units without converting them first?
5. How can rounding affect the accuracy of a result in scientific notation?

**Tip:** Always count significant figures from the least precise measurement when multiplying or dividing values.

Express this product in units of km^3 in standard scientific notation and with the appropriate number of significant figures: (3.4 km) × (4.0 m) × (13.24 × 10−3 mm)

Learn how to convert and multiply physical quantities in different units to express the result in cubic kilometers (km³). This problem covers unit conversion from meters and millimeters to kilometers, multiplication of quantities, and expressing the final result in scientific notation with significant figures. Suitable for students in Grades 9-11.

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