Multiply. Write Your Answer In Scientific Notation.$\[ 6 \cdot \left(1 \times 10^2\right) \\]$\[\square\\]
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Understanding Scientific Notation
Scientific notation is a way of expressing very large or very small numbers in a compact form. It consists of a number between 1 and 10, multiplied by a power of 10. This notation is commonly used in mathematics, physics, and engineering to simplify calculations and express numbers in a more manageable way.
Example of Scientific Notation
1 × 10^2
In this example, the number 1 is multiplied by 10 raised to the power of 2. This can be read as "1 times 10 to the power of 2" or simply "10 squared". The result is 100, which is a large number that can be easily expressed in scientific notation.
Multiplying Numbers in Scientific Notation
When multiplying numbers in scientific notation, we need to follow a specific set of rules to ensure that the result is also in scientific notation. The rules are as follows:
- Multiply the numbers as if they were in standard form (i.e., without the powers of 10).
- Add the exponents of the powers of 10.
- Write the result in scientific notation, using the product of the numbers as the coefficient and the sum of the exponents as the power of 10.
Example: 6 × (1 × 10^2)
Step 1: Multiply the Numbers
First, we multiply the numbers as if they were in standard form. In this case, we have 6 multiplied by 100 (which is the result of 1 × 10^2).
6 × 100 = 600
Step 2: Add the Exponents
Next, we add the exponents of the powers of 10. In this case, we have 10^2 (which is 100) multiplied by 10^0 (which is 1).
10^2 + 10^0 = 10^2 + 1 = 10^2
Step 3: Write the Result in Scientific Notation
Finally, we write the result in scientific notation, using the product of the numbers as the coefficient and the sum of the exponents as the power of 10.
6 × (1 × 10^2) = 6 × 10^2 = 600 × 10^0 = 6 × 10^2
The result is 6 × 10^2, which is equal to 600.
Conclusion
Multiplying numbers in scientific notation is a straightforward process that involves multiplying the numbers as if they were in standard form, adding the exponents of the powers of 10, and writing the result in scientific notation. By following these simple rules, we can easily multiply numbers in scientific notation and express the result in a compact and manageable form.
Key Takeaways
1. Multiply the numbers as if they were in standard form.
2. Add the exponents of the powers of 10.
3. Write the result in scientific notation, using the product of the numbers as the coefficient and the sum of the exponents as the power of 10.
By mastering the art of multiplying numbers in scientific notation, we can simplify complex calculations and express numbers in a more intuitive and manageable way. Whether you're working in mathematics, physics, or engineering, scientific notation is an essential tool that can help you solve problems more efficiently and effectively.
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Frequently Asked Questions
Q: What is scientific notation?
A: Scientific notation is a way of expressing very large or very small numbers in a compact form. It consists of a number between 1 and 10, multiplied by a power of 10.
Q: How do I multiply numbers in scientific notation?
A: To multiply numbers in scientific notation, you need to follow these steps:
- Multiply the numbers as if they were in standard form (i.e., without the powers of 10).
- Add the exponents of the powers of 10.
- Write the result in scientific notation, using the product of the numbers as the coefficient and the sum of the exponents as the power of 10.
Q: What if the exponents are different?
A: If the exponents are different, you need to add them together. For example, if you have 3 × 10^4 and 2 × 10^3, you would add the exponents: 4 + 3 = 7. The result would be 6 × 10^7.
Q: Can I multiply numbers with negative exponents?
A: Yes, you can multiply numbers with negative exponents. To do this, you need to follow the same steps as before, but with the negative exponent. For example, if you have 2 × 10^-2 and 3 × 10^-3, you would add the exponents: -2 + (-3) = -5. The result would be 6 × 10^-5.
Q: How do I divide numbers in scientific notation?
A: To divide numbers in scientific notation, you need to follow these steps:
- Divide the numbers as if they were in standard form (i.e., without the powers of 10).
- Subtract the exponents of the powers of 10.
- Write the result in scientific notation, using the quotient of the numbers as the coefficient and the difference of the exponents as the power of 10.
Q: What if I have a zero exponent?
A: If you have a zero exponent, the result is always 1. For example, if you have 2 × 10^0, the result is simply 2.
Q: Can I multiply or divide numbers with different bases?
A: No, you cannot multiply or divide numbers with different bases. The bases must be the same for both numbers. For example, you cannot multiply 2 × 10^2 and 3 × 10^3, because the bases are different.
Q: How do I convert a number from standard form to scientific notation?
A: To convert a number from standard form to scientific notation, you need to follow these steps:
- Move the decimal point to the left or right until you have a number between 1 and 10.
- Multiply the number by 10 raised to the power of the number of places you moved the decimal point.
Q: Can I use scientific notation with fractions?
A: Yes, you can use scientific notation with fractions. To do this, you need to follow the same steps as before, but with the fraction. For example, if you have 1/2 × 10^2, you would multiply the fraction by 10^2: 1/2 × 10^2 = 5 × 10^1.
Conclusion
Multiplying numbers in scientific notation can seem daunting at first, but with practice and patience, you can master this skill. Remember to follow the steps outlined above, and you'll be able to multiply numbers in scientific notation with ease. Whether you're working in mathematics, physics, or engineering, scientific notation is an essential tool that can help you solve problems more efficiently and effectively.
Key Takeaways
1. Multiply the numbers as if they were in standard form.
2. Add the exponents of the powers of 10.
3. Write the result in scientific notation, using the product of the numbers as the coefficient and the sum of the exponents as the power of 10.
By mastering the art of multiplying numbers in scientific notation, you can simplify complex calculations and express numbers in a more intuitive and manageable way.