Question

Vocabulary Terms - Define/Illustrate

Exponent Rule for dividing like bases applied to SN:
What words tell you to divide:
Why is “how many times larger” a division problem?

Problems - SHOW ALL WORK!

1. (6.3 * 10^8)/(2.1 * 10^3) =

2. (9.4 * 10^4)/(4.7 * 10^7)

3. In 2010, there were about 4 x 10^7 people living in California and about 1 x 10^6 people living in Rhode Island.
How many times larger was the population of California than the population of Rhode Island?

4. Pablo divided 6 x 10^8 by 2 x 10^-4, and got a final answer of 3 x 10^-2.

Explain what mistake Pablo made, and give the correct solution.

Write a division problem in which two numbers written in scientific notation have a quotient of 5 x 10^-5.






Show how you would solve it.

Answer 1

### Vocabulary Terms

**Exponent Rule for Dividing Like Bases Applied to Scientific Notation:**
When dividing two numbers that are written in scientific notation, you can apply the exponent rule for division which states that when dividing like bases (in this case, the base 10), you should subtract the exponents. In mathematical terms:
\[
\frac{a \times 10^m}{b \times 10^n} = \frac{a}{b} \times 10^{m-n}
\]

**Words That Tell You to Divide:**
- "How many times larger"
- "Per"
- "Out of"
- "Every"
- "Rate of"

**Why is “How Many Times Larger” a Division Problem?**
The phrase "how many times larger" implies a comparison between two quantities. To find how many times larger one quantity is than another, you divide the larger quantity by the smaller one. This quantifies the difference in size or value between the two quantities.

### Problems

1. **Problem:** \((6.3 \times 10^8)/(2.1 \times 10^3)\)

**Solution:**
\[
\frac{6.3}{2.1} = 3
\]
\[
10^{8-3} = 10^{5}
\]
Combining these results:
\[
3 \times 10^{5}
\]
Thus,
\[
\frac{6.3 \times 10^8}{2.1 \times 10^3} = 3.0 \times 10^5
\]

2. **Problem:** \((9.4 \times 10^4)/(4.7 \times 10^7)\)

**Solution:**
\[
\frac{9.4}{4.7} = 2
\]
\[
10^{4-7} = 10^{-3}
\]
Combining these results:
\[
2 \times 10^{-3}
\]
Thus,
\[
\frac{9.4 \times 10^4}{4.7 \times 10^7} = 2.0 \times 10^{-3}
\]

3. **Problem:** In 2010, there were about \(4 \times 10^7\) people living in California and about \(1 \times 10^6\) people living in Rhode Island. How many times larger was the population of California than the population of Rhode Island?

**Solution:**
\[
\frac{4 \times 10^7}{1 \times 10^6} = \frac{4}{1} \times 10^{7-6} = 4 \times 10^{1} = 40
\]
Thus, the population of California was **40 times larger** than the population of Rhode Island.

4. **Problem:** Pablo divided \(6 \times 10^8\) by \(2 \times 10^{-4}\), and got a final answer of \(3 \times 10^{-2}\). Explain what mistake Pablo made, and give the correct solution.

**Pablo's Mistake:**
The mistake Pablo made was in the calculation of the exponent when dividing. He should have subtracted \(-4\) from \(8\) instead of incorrectly calculating or misinterpreting the result.

**Correct Solution:**
\[
\frac{6 \times 10^8}{2 \times 10^{-4}} = \frac{6}{2} \times 10^{8 - (-4)} = 3 \times 10^{8 + 4} = 3 \times 10^{12}
\]

Thus, the correct answer is \(3 \times 10^{12}\).

5. **Write a division problem in which two numbers written in scientific notation have a quotient of \(5 \times 10^{-5}\).**

Let's take:
- Numerator: \(1 \times 10^{-4}\)
- Denominator: \(2 \times 10^{1}\)

**Division Problem:**
\[
\frac{1 \times 10^{-4}}{2 \times 10^{1}}
\]

**Solution:**
\[
= \frac{1}{2} \times 10^{-4 - 1} = 0.5 \times 10^{-5} = 5 \times 10^{-6}
\]
To achieve exactly \(5 \times 10^{-5}\), we can adjust as follows:

- Numerator: \(1 \times 10^{-4} \times 10^{1} = 1 \times 10^{-3}\)
- Denominator: \(2 \times 10^{1} \times 10^{1} = 2 \times 10^{2}\)

Thus, the expression:
\[
\frac{1 \times 10^{-4}}{2 \times 10^{-5}} = 5 \times 10^{-5}
\]

### Summary
The division of scientific notation involves applying the exponent rules and careful calculation to find accurate comparisons and ratios within scientific contexts.