Find The Product And Express Your Answer In Scientific Notation. 15 , 000 , 000 × ( 3.5 × 10 − 7 15,000,000 \times (3.5 \times 10^{-7} 15 , 000 , 000 × ( 3.5 × 1 0 − 7 ]Product: □ \square □

What is Scientific Notation?

Scientific notation is a way of expressing very large or very small numbers in a more manageable and concise form. It is a method of writing numbers as a product of a number between 1 and 10 and a power of 10. This notation is commonly used in mathematics, physics, engineering, and other fields where large or small numbers are encountered frequently.

The Product: $15,000,000 \times (3.5 \times 10^{-7})$

In this problem, we are given the product of two numbers: $15,000,000$ and $(3.5 \times 10^{-7})$. Our task is to find the product of these two numbers and express the result in scientific notation.

Step 1: Multiply the Numbers

To find the product, we can simply multiply the two numbers together. However, we need to be careful when multiplying numbers in scientific notation. We need to multiply the coefficients (the numbers in front of the powers of 10) and add the exponents of the powers of 10.

Multiplying the Coefficients

The coefficient of the first number is $15,000,000$, and the coefficient of the second number is $3.5$. To multiply these coefficients, we can simply multiply them together:

$15,000,000 \times 3.5 = 52,500,000$

Adding the Exponents

The exponent of the first number is $0$ (since it is a whole number), and the exponent of the second number is $-7$. To add the exponents, we can simply add them together:

$0 + (-7) = -7$

The Product in Scientific Notation

Now that we have multiplied the coefficients and added the exponents, we can write the product in scientific notation:

$52,500,000 \times 10^{-7} = 5.25 \times 10^{11} \times 10^{-7}$

Simplifying the Product

To simplify the product, we can combine the powers of 10 by adding their exponents:

$5.25 \times 10^{11} \times 10^{-7} = 5.25 \times 10^{11-7}$

$5.25 \times 10^{11-7} = 5.25 \times 10^{4}$

The Final Answer

Therefore, the product of $15,000,000$ and $(3.5 \times 10^{-7})$ is:

$5.25 \times 10^{4}$

Conclusion

In this problem, we used scientific notation to express a large number in a more manageable form. We multiplied two numbers together and expressed the result in scientific notation. We also simplified the product by combining the powers of 10. This problem demonstrates the power of scientific notation in expressing large and small numbers.

Real-World Applications

Scientific notation is used in many real-world applications, such as:

• Physics: Scientific notation is used to express large and small numbers in physics, such as the speed of light (approximately $3.00 \times 10^{8}$ meters per second) and the Planck constant (approximately $6.626 \times 10^{-34}$ joule-seconds).
• Engineering: Scientific notation is used to express large and small numbers in engineering, such as the size of electronic components (e.g., $10^{-6}$ meters) and the strength of materials (e.g., $10^{9}$ pascals).
• Computer Science: Scientific notation is used to express large and small numbers in computer science, such as the size of data structures (e.g., $10^{12}$ bytes) and the speed of algorithms (e.g., $10^{9}$ operations per second).

Common Misconceptions

There are several common misconceptions about scientific notation:

• Misconception 1: Scientific notation is only used for very large numbers. Reality: Scientific notation can be used for both very large and very small numbers.
• Misconception 2: Scientific notation is only used in mathematics and physics. Reality: Scientific notation is used in many fields, including engineering, computer science, and economics.
• Misconception 3: Scientific notation is difficult to understand. Reality: Scientific notation is a simple and powerful tool for expressing large and small numbers.

Conclusion

Q: What is scientific notation?

A: Scientific notation is a way of expressing very large or very small numbers in a more manageable and concise form. It is a method of writing numbers as a product of a number between 1 and 10 and a power of 10.

Q: How do I write a number in scientific notation?

A: To write a number in scientific notation, you need to express it as a product of a number between 1 and 10 and a power of 10. For example, the number 456,789 can be written in scientific notation as 4.56789 × 10^5.

Q: What is the exponent in scientific notation?

A: The exponent in scientific notation is the power of 10 that is multiplied by the coefficient (the number between 1 and 10). For example, in the number 4.56789 × 10^5, the exponent is 5.

Q: How do I multiply numbers in scientific notation?

A: To multiply numbers in scientific notation, you need to multiply the coefficients (the numbers between 1 and 10) and add the exponents of the powers of 10. For example, to multiply 3.45 × 10^2 and 2.75 × 10^3, you would multiply the coefficients (3.45 and 2.75) and add the exponents (2 and 3) to get 9.4875 × 10^5.

Q: How do I divide numbers in scientific notation?

A: To divide numbers in scientific notation, you need to divide the coefficients (the numbers between 1 and 10) and subtract the exponents of the powers of 10. For example, to divide 4.56789 × 10^5 by 2.75 × 10^3, you would divide the coefficients (4.56789 and 2.75) and subtract the exponents (5 and 3) to get 1.665 × 10^2.

Q: What is the difference between scientific notation and standard notation?

A: Scientific notation and standard notation are two different ways of expressing numbers. Standard notation is the way we normally write numbers, with all the digits in a row. Scientific notation, on the other hand, is a way of expressing numbers as a product of a number between 1 and 10 and a power of 10.

Q: When should I use scientific notation?

A: You should use scientific notation when you need to express very large or very small numbers in a more manageable and concise form. For example, if you are working with numbers that are too large or too small to be easily expressed in standard notation, scientific notation can be a useful tool.

Q: What are some common mistakes to avoid when using scientific notation?

A: Some common mistakes to avoid when using scientific notation include:

• Not following the rules for multiplying and dividing numbers in scientific notation
• Not using the correct exponent
• Not expressing the number in the correct form (e.g., 4.56789 × 10^5 instead of 456,789)

Q: How can I practice using scientific notation?

A: You can practice using scientific notation by working with numbers that are too large or too small to be easily expressed in standard notation. You can also try converting numbers from standard notation to scientific notation and vice versa.

Q: What are some real-world applications of scientific notation?

A: Scientific notation has many real-world applications, including:

• Physics: Scientific notation is used to express large and small numbers in physics, such as the speed of light (approximately 3.00 × 10^8 meters per second) and the Planck constant (approximately 6.626 × 10^-34 joule-seconds).
• Engineering: Scientific notation is used to express large and small numbers in engineering, such as the size of electronic components (e.g., 10^-6 meters) and the strength of materials (e.g., 10^9 pascals).
• Computer Science: Scientific notation is used to express large and small numbers in computer science, such as the size of data structures (e.g., 10^12 bytes) and the speed of algorithms (e.g., 10^9 operations per second).

Conclusion

In conclusion, scientific notation is a powerful tool for expressing large and small numbers in a more manageable and concise form. By understanding how to use scientific notation, you can better appreciate the beauty and complexity of the world around us.