Question

Vocabulary Terms - Define/Illustrate

Adding/Subtracting Fractions - we need a _______________

Example:
Adding/Subtracting expressions - we need
______________

Example:
Adding/Subtracting in SN - we need
______________

Example:


5 Problems - SHOW ALL WORK! Write all answers in Scientific Notation.

In 2012, the population of NY was about 2.4 x 10^7 and the population of Oregon was 4 x 10^5.
What is the population of the two states combined?





3.71 x 10^5 + 4.65 x 10^6

7.35 x 10^-7 + 2.16 x 10^-8








5 x 10^3 yd^2 is the measurement of your yard and 3 x 10^5 yd^2 is the measurement of your neighbor’s yard.
Whose yard is bigger?


If you fenced in both yards together, how big would the yard be?



In 2006, the US had a Gross Domestic Product (GDP) of 1.3 x 10^13. The United Kingdom had a GDP of 1.9 x 10^12.
What were the combined GDPs of the US and the UK in 2006?


Laila’s Solution
Identify and Explain the error
(1.313 x 10^13) + (1.93 x 10^12)
1.313 + 1.93 = 3.243
10^13 + 10^12 = 10^25

Answer 1

Sure! Let's fill in the blanks and solve the problems step by step.

### Vocabulary Terms

1. **Adding/Subtracting Fractions - we need a** **common denominator.**

2. **Adding/Subtracting expressions - we need** **like terms.**

3. **Adding/Subtracting in SN - we need** **the same exponent.**

### Example Problems

1. **In 2012, the population of NY was about 2.4 x 10^7 and the population of Oregon was 4 x 10^5. What is the population of the two states combined?**

The populations are:
- New York: \( 2.4 \times 10^7 \)
- Oregon: \( 4 \times 10^5 \)

**Step 1**: Convert Oregon's population to match New York's exponent:
\[
4 \times 10^5 = 0.04 \times 10^7
\]

**Step 2**: Now add the populations together:
\[
2.4 \times 10^7 + 0.04 \times 10^7 = (2.4 + 0.04) \times 10^7 = 2.44 \times 10^7
\]

**Answer:** \( 2.44 \times 10^7 \)

2. **3.71 x 10^5 + 4.65 x 10^6**

**Step 1**: Convert \( 4.65 \times 10^6 \) to have the same exponent:
\[
4.65 \times 10^6 = 46.5 \times 10^5
\]

**Step 2**: Now add the values:
\[
3.71 \times 10^5 + 46.5 \times 10^5 = (3.71 + 46.5) \times 10^5 = 50.21 \times 10^5
\]

**Step 3**: Convert \( 50.21 \times 10^5 \) to scientific notation:
\[
50.21 \times 10^5 = 5.021 \times 10^6
\]

**Answer:** \( 5.021 \times 10^6 \)

3. **7.35 x 10^-7 + 2.16 x 10^-8**

**Step 1**: Convert \( 2.16 \times 10^{-8} \) to \( 10^{-7} \):
\[
2.16 \times 10^{-8} = 0.216 \times 10^{-7}
\]

**Step 2**: Now add the two values:
\[
7.35 \times 10^{-7} + 0.216 \times 10^{-7} = (7.35 + 0.216) \times 10^{-7} = 7.566 \times 10^{-7}
\]

**Answer:** \( 7.566 \times 10^{-7} \)

4. **5 x 10^3 yd^2 is the measurement of your yard and 3 x 10^5 yd^2 is the measurement of your neighbor’s yard. Whose yard is bigger?**

Compare:
- Your yard: \( 5 \times 10^3 = 5000 \, yd^2 \)
- Neighbor's yard: \( 3 \times 10^5 = 300000 \, yd^2 \)

**Answer:** Your neighbor's yard is bigger.

5. **If you fenced in both yards together, how big would the yard be?**

**Step 1**: Convert \( 5 \times 10^3 \) to the same exponent as \( 3 \times 10^5 \):
\[
5 \times 10^3 = 0.05 \times 10^5
\]

**Step 2**: Now, add the areas:
\[
0.05 \times 10^5 + 3 \times 10^5 = (0.05 + 3) \times 10^5 = 3.05 \times 10^5 \, yd^2
\]

**Answer:** \( 3.05 \times 10^5 \, yd^2 \)

6. **In 2006, the US had a GDP of 1.3 x 10^13. The United Kingdom had a GDP of 1.9 x 10^12. What were the combined GDPs of the US and the UK in 2006?**

**Step 1**: Convert \( 1.9 \times 10^{12} \) to have the same exponent:
\[
1.9 \times 10^{12} = 0.19 \times 10^{13}
\]

**Step 2**: Now add the GDPs:
\[
1.3 \times 10^{13} + 0.19 \times 10^{13} = (1.3 + 0.19) \times 10^{13} = 1.49 \times 10^{13}
\]

**Answer:** \( 1.49 \times 10^{13} \)

7. **Laila’s Solution - Identify and Explain the error:**
- Laila wrote:
\[
(1.313 \times 10^{13}) + (1.93 \times 10^{12})
\]
- She attempted to add the mantissas incorrectly by neglecting the exponent equivalency:
- Correct approach: \( 1.93 \times 10^{12} = 0.193 \times 10^{13} \).
- Properly adding yields:
\[
1.313 \times 10^{13} + 0.193 \times 10^{13} = (1.313 + 0.193) \times 10^{13} = 1.506 \times 10^{13}
\]

**Error**: She erroneously added different powers of 10, mistakenly treating \( 10^{13} \) and \( 10^{12} \) as the same.

These are the definitions, step-by-step solutions, and explanations of the errors for the provided problems. If you have any more questions or need further assistance, feel free to ask!