a kilometer is how many times large as a micrometer? show work using powers of 10 to calculate answer

To calculate how many times larger a kilometer is compared to a micrometer, we can express both units in terms of powers of 10 and then divide them.

### Step 1: Express the units in powers of 10
- **1 kilometer (km)** is equal to \( 1 \times 10^3 \) meters. 
- **1 micrometer (μm)** is equal to \( 1 \times 10^{-6} \) meters.

### Step 2: Divide the two quantities
Now, to find how many times larger a kilometer is compared to a micrometer, divide 1 kilometer by 1 micrometer:

\[
\frac{1 \text{ km}}{1 \text{ μm}} = \frac{1 \times 10^3 \text{ meters}}{1 \times 10^{-6} \text{ meters}} = \frac{10^3}{10^{-6}}
\]

### Step 3: Simplify the powers of 10
Using the laws of exponents, when dividing powers of 10, we subtract the exponents:

\[
10^3 \div 10^{-6} = 10^{3 - (-6)} = 10^{3 + 6} = 10^9
\]

### Final Answer:
A kilometer is \( 10^9 \) times larger than a micrometer, or 1 billion times larger.

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Would you like further details or clarifications?

Here are some related questions to expand your understanding:
1. How many micrometers are in one meter?
2. How does the conversion between millimeters and micrometers work using powers of 10?
3. How many kilometers are equivalent to one nanometer?
4. Can you convert 5 kilometers to micrometers using powers of 10?
5. What is the relationship between a kilometer and a millimeter in terms of powers of 10?

**Tip**: When converting between units, always break them down into meters (the base unit) using powers of 10, as it simplifies calculations significantly.

Learn how to calculate how many times larger a kilometer is compared to a micrometer using powers of 10. This step-by-step solution covers unit conversion and exponentiation, demonstrating that a kilometer is 1 billion (10^9) times larger than a micrometer. Ideal for students in Grades 6-8.

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