Question
EXAMPLE 2 Find the quotient: \( \frac{2.4 \times 10^{7}}{9.6 \times 10^{3}} \). Divide the multipliers: \( 2.4 \div 9.6=0.25 \) Divide the powers of ten: \( \frac{10^{7}}{10^{3}}=10^{7-3}=10^{4} \) Combine the answers and write the product in scientific notation. \( 0.25 \times 10^{4}=0.25 \times\left(10 \times 10^{3}\right)=(0.25 \times 10) \times 10^{3}=2.5 \times 10^{3} \) EXERCISES \begin{tabular}{ll} Write each number in scientific notation. \( ( \) Lessons \( 2.2,2.3) \) \\ 1. \( 25,500,000 \) \\ Write each number in standard notation. \( ( \) Lessons \( 2.2,2.3) \) \\ 3. \( 5.23 \times 10^{4} \) \\ Simplify each expression. \( ( \) Lessons \( 2.1,2.4) \) \\ 5. \( (9-7)^{3} \cdot 5^{0}+(8+3)^{2} \) \\ 7. \( 3.2 \times 10^{5}+1.25 \times 10^{4}+2.9 \times 10^{5} \) & 4. \( 1.33 \times 10^{-5} \) \\ \hline\end{tabular}
Ask by Perkins Martin. in the United States
Mar 12,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
1. \( 25,500,000 = 2.55 \times 10^{7} \)
2. \( 5.23 \times 10^{4} = 52,300 \)
3. \( (9-7)^{3} \cdot 5^{0} + (8+3)^{2} = 129 \)
4. \( 3.2 \times 10^{5} + 1.25 \times 10^{4} + 2.9 \times 10^{5} = 6.225 \times 10^{5} \)
Solution
Let's solve the exercises step by step.
### Exercise 1: Write \( 25,500,000 \) in scientific notation.
To convert a number into scientific notation, we need to express it in the form \( a \times 10^n \), where \( 1 \leq a < 10 \) and \( n \) is an integer.
1. **Identify the significant figures**: The number \( 25,500,000 \) can be expressed as \( 2.55 \) (keeping two significant figures).
2. **Count the decimal places moved**: The decimal point in \( 2.55 \) is moved 7 places to the right to get back to \( 25,500,000 \).
3. **Write in scientific notation**: Thus, we have:
\[
25,500,000 = 2.55 \times 10^{7}
\]
### Exercise 2: Write \( 5.23 \times 10^{4} \) in standard notation.
To convert from scientific notation to standard notation, we move the decimal point to the right by the exponent's value.
1. **Move the decimal point**: Move the decimal point in \( 5.23 \) four places to the right:
- \( 5.23 \) becomes \( 52300 \) after moving the decimal point four places.
2. **Write in standard notation**: Thus, we have:
\[
5.23 \times 10^{4} = 52300
\]
### Exercise 3: Simplify \( (9-7)^{3} \cdot 5^{0}+(8+3)^{2} \).
1. **Calculate \( (9-7)^{3} \)**:
\[
(9-7)^{3} = 2^{3} = 8
\]
2. **Calculate \( 5^{0} \)**:
\[
5^{0} = 1
\]
3. **Combine the results**:
\[
(9-7)^{3} \cdot 5^{0} = 8 \cdot 1 = 8
\]
4. **Calculate \( (8+3)^{2} \)**:
\[
(8+3)^{2} = 11^{2} = 121
\]
5. **Final calculation**:
\[
8 + 121 = 129
\]
### Exercise 4: Simplify \( 3.2 \times 10^{5}+1.25 \times 10^{4}+2.9 \times 10^{5} \).
1. **Align the powers of ten**: Convert \( 1.25 \times 10^{4} \) to the same power as the others:
\[
1.25 \times 10^{4} = 0.125 \times 10^{5}
\]
2. **Combine the terms**:
\[
3.2 \times 10^{5} + 0.125 \times 10^{5} + 2.9 \times 10^{5} = (3.2 + 0.125 + 2.9) \times 10^{5}
\]
3. **Calculate the sum**:
\[
3.2 + 0.125 + 2.9 = 6.225
\]
4. **Final result**:
\[
3.2 \times 10^{5} + 1.25 \times 10^{4} + 2.9 \times 10^{5} = 6.225 \times 10^{5}
\]
### Summary of Answers:
1. \( 25,500,000 = 2.55 \times 10^{7} \)
2. \( 5.23 \times 10^{4} = 52300 \)
3. \( (9-7)^{3} \cdot 5^{0}+(8+3)^{2} = 129 \)
4. \( 3.2 \times 10^{5}+1.25 \times 10^{4}+2.9 \times 10^{5} = 6.225 \times 10^{5} \)
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Did you know that scientific notation is a game-changer in the world of mathematics and science? It allows us to easily handle incredibly large or small numbers, like the distance from Earth to a star or the size of a cell! For instance, instead of writing 0.0000000001 (a billionth), you just write \( 1 \times 10^{-10} \). This makes calculations and comparisons so much simpler and neater! If you're converting numbers to and from scientific notation, a common pitfall is misplacing the decimal point. When shifting from standard to scientific, the decimal moves to create a number between 1 and 10; the count of shifts becomes the exponent on 10. Failing to get your decimal right can lead you astray. Always double-check to ensure those digits are chilling in their proper place!
