Question

Vocabulary Terms - Define/Illustrate

Exponent Rule for Multiplication of like bases:
Words that tell you to multiply:
What happens when you multiply by a large number?

What happens when you multiply by a small number?

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

1. (5.4 * 10^-1)(7 * 10^0) =
2. (5 * 10^3)(3.5 * 10^-1) =
3. Statement: The distance from the Earth to the Sun is about 400 times the distance from Earth to the Moon.
Shortest Distance from Earth
Moon - 2.3808 * 10^5 mi
Sun - 9.3 * 10^7
Is this true?


Why?

4. If the speed of radio waves is
1.86 x 10^5 miles per second
How much time would it take to send a text message to the Moon in minutes?

Answer 1

### Vocabulary Terms - Definitions/Illustrations

**Exponent Rule for Multiplication of Like Bases:**
When multiplying two numbers with the same base, you add their exponents. This rule is stated mathematically as:
\[
a^m \cdot a^n = a^{m+n}
\]

**Words that tell you to multiply:**
Words such as "times," "multiplied by," "product," and "of" indicate multiplication. For instance, "3 times 4" or "the product of 5 and 6."

**What happens when you multiply by a large number?**
When you multiply by a large number, the product increases significantly. If the base number is in scientific notation, the exponent can also increase, representing a larger value.

**What happens when you multiply by a small number?**
When you multiply by a small number (less than 1), the product becomes smaller. For base numbers in scientific notation, the exponent's absolute value may increase, indicating a smaller overall number.

### Problems - Show All Work! Write All Answers in Scientific Notation.

1. \((5.4 \times 10^{-1})(7 \times 10^{0})\)

Step 1: Multiply the coefficients: \(5.4 \times 7 = 37.8\).

Step 2: Multiply the powers of ten: \(10^{-1} \times 10^{0} = 10^{-1 + 0} = 10^{-1}\).

Step 3: Combine results:

\[
37.8 \times 10^{-1}
\]

Step 4: Convert to scientific notation:

\[
3.78 \times 10^{0}
\]

**Final Answer:** \(3.78 \times 10^{0}\)


2. \((5 \times 10^{3})(3.5 \times 10^{-1})\)

Step 1: Multiply the coefficients: \(5 \times 3.5 = 17.5\).

Step 2: Multiply the powers of ten: \(10^{3} \times 10^{-1} = 10^{3 - 1} = 10^{2}\).

Step 3: Combine results:

\[
17.5 \times 10^{2}
\]

Step 4: Convert to scientific notation:

\[
1.75 \times 10^{3}
\]

**Final Answer:** \(1.75 \times 10^{3}\)


3. Statement: The distance from the Earth to the Sun is about 400 times the distance from Earth to the Moon.

- Shortest Distance from Earth to Moon: \(2.3808 \times 10^{5}\) miles
- Shortest Distance from Earth to Sun: \(9.3 \times 10^{7}\) miles

**Calculating the ratio:**

Find \(400 \times (2.3808 \times 10^{5})\):
\[
400 = 4.0 \times 10^{2} \implies 4.0 \times 10^{2} \times 2.3808 \times 10^{5}
\]
Step 1: Multiply the coefficients: \(4.0 \times 2.3808 = 9.5232\).

Step 2: Multiply the powers of ten: \(10^{2} \times 10^{5} = 10^{7}\).

Step 3: Combine results:
\[
9.5232 \times 10^{7}
\]

- Comparing \(9.30 \times 10^{7}\) with \(9.5232 \times 10^{7}\):

**Conclusion:**
The statement is approximately true because \(9.5232 \times 10^{7}\) is indeed a bit more than \(9.3 \times 10^{7}\), which can reasonably be approximated as 400 times.


4. If the speed of radio waves is \(1.86 \times 10^{5}\) miles per second, to find the time it takes to send a text message to the Moon:

- Distance to the Moon: \(2.3808 \times 10^{5}\) miles
- Time (in seconds) = Distance / Speed = \(\frac{2.3808 \times 10^{5}}{1.86 \times 10^{5}}\)

Step 1: Divide the coefficients:
\[
\frac{2.3808}{1.86} \approx 1.2786
\]

Step 2: Divide the powers of ten:
\[
\frac{10^{5}}{10^{5}} = 10^{0} = 1
\]

Time (in seconds) is approximately \(1.2786 \text{ seconds}\).

Step 3: Convert from seconds to minutes. (There are 60 seconds in a minute):
\[
\text{Time in minutes} = \frac{1.2786}{60} \approx 0.0213 \text{ minutes}
\]

**Final Answer:** Approximately \(0.0213\) minutes or \(1.2786\) seconds.