What Is The Quotient Of 1, Point, 0, 6, Times, 10, To The Power 71.06×10 7 And 2, Point, 6, 5, Times, 10, To The Power 42.65×10 4 Expressed In Scientific Notation?

Feb 28, 2025 by ADMIN 164 views

What is the Quotient of 1.0.6 × 10^71.06×107 and 2.6.5 × 10^42.65×104 Expressed in Scientific Notation?

Understanding the Problem

The problem requires us to find the quotient of two very large numbers expressed in scientific notation. The first number is 1.0.6 × 10^71.06×107 and the second number is 2.6.5 × 10^42.65×104. To find the quotient, we need to first simplify the exponents and then divide the coefficients.

Simplifying the Exponents

The first exponent is 71.06×10^7. To simplify this, we need to multiply 71.06 by 10^7. This can be done by adding the exponents, which gives us 71.06×10^7 = 71.06 × 10^(7+1) = 71.06 × 10^8.

The second exponent is 42.65×10^4. To simplify this, we need to multiply 42.65 by 10^4. This can be done by adding the exponents, which gives us 42.65×10^4 = 42.65 × 10^(4+1) = 42.65 × 10^5.

Simplifying the Numbers

Now that we have simplified the exponents, we can simplify the numbers. The first number is 1.0.6 × 10^71.06×107, which is equal to 1.06 × 10^8. The second number is 2.6.5 × 10^42.65×104, which is equal to 26.5 × 10^5.

Finding the Quotient

Now that we have simplified the numbers, we can find the quotient. To do this, we need to divide the coefficients and subtract the exponents. The quotient is (1.06 × 10^8) / (26.5 × 10^5).

Performing the Division

To perform the division, we need to divide the coefficients and subtract the exponents. The quotient is (1.06 / 26.5) × 10^(8-5).

Calculating the Quotient

Now that we have the quotient, we can calculate the value. The quotient is (1.06 / 26.5) × 10^3.

Evaluating the Quotient

To evaluate the quotient, we need to divide 1.06 by 26.5. This gives us a quotient of approximately 0.04.

Expressing the Quotient in Scientific Notation

The quotient is approximately 0.04 × 10^3. To express this in scientific notation, we need to move the decimal point three places to the right. This gives us 4 × 10^(-2).

Conclusion

In conclusion, the quotient of 1.0.6 × 10^71.06×107 and 2.6.5 × 10^42.65×104 expressed in scientific notation is 4 × 10^(-2).

Understanding Scientific Notation

Scientific notation is a way of expressing very large or very small numbers in a more manageable form. It consists of a coefficient and an exponent of 10. The coefficient is a number between 1 and 10, and the exponent is a power of 10.

Examples of Scientific Notation

Some examples of scientific notation include:

1.0.6 × 10^71.06×107
2.6.5 × 10^42.65×104
4 × 10^(-2)
3.14 × 10^1

Advantages of Scientific Notation

Scientific notation has several advantages. It allows us to express very large or very small numbers in a more manageable form, making it easier to perform calculations. It also makes it easier to compare numbers and to perform operations such as addition and subtraction.

Disadvantages of Scientific Notation

Scientific notation also has some disadvantages. It can be difficult to understand and use, especially for those who are not familiar with it. It can also be difficult to convert numbers from scientific notation to standard notation.

Real-World Applications of Scientific Notation

Scientific notation has many real-world applications. It is used in fields such as physics, chemistry, and engineering to express very large or very small numbers. It is also used in finance and economics to express very large or very small amounts of money.

Conclusion

In conclusion, scientific notation is a powerful tool for expressing very large or very small numbers in a more manageable form. It has many advantages, including making it easier to perform calculations and compare numbers. However, it also has some disadvantages, including being difficult to understand and use. Despite these disadvantages, scientific notation is an essential tool in many fields and is widely used in real-world applications.
Frequently Asked Questions (FAQs) About Scientific Notation

Q: What is scientific notation?

A: Scientific notation is a way of expressing very large or very small numbers in a more manageable form. It consists of a coefficient and an exponent 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 number between 1 and 10 multiplied by 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 coefficient in scientific notation?

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

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. For example, in the number 4.56789 × 10^5, the exponent is 5.

Q: How do I convert a number from scientific notation to standard notation?

A: To convert a number from scientific notation to standard notation, you need to multiply the coefficient by the power of 10. For example, the number 4.56789 × 10^5 can be converted to standard notation by multiplying 4.56789 by 10^5, which gives 456,789.

Q: How do I add or subtract numbers in scientific notation?

A: To add or subtract numbers in scientific notation, you need to add or subtract the coefficients and then add or subtract the exponents. For example, the numbers 4.56789 × 10^5 and 2.34567 × 10^5 can be added by adding the coefficients (4.56789 + 2.34567 = 6.91356) and then adding the exponents (5 + 5 = 10), which gives 6.91356 × 10^10.

Q: How do I multiply or divide numbers in scientific notation?

A: To multiply or divide numbers in scientific notation, you need to multiply or divide the coefficients and then add or subtract the exponents. For example, the numbers 4.56789 × 10^5 and 2.34567 × 10^5 can be multiplied by multiplying the coefficients (4.56789 × 2.34567 = 10.73751) and then adding the exponents (5 + 5 = 10), which gives 10.73751 × 10^10.

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

A: Some common mistakes to avoid when working with scientific notation include:

Forgetting to include the exponent
Forgetting to include the coefficient
Adding or subtracting the exponents incorrectly
Multiplying or dividing the coefficients incorrectly
Not converting the number to standard notation when necessary

Q: Why is scientific notation important?

A: Scientific notation is important because it allows us to express very large or very small numbers in a more manageable form. This makes it easier to perform calculations and compare numbers.

Q: When should I use scientific notation?

A: You should use scientific notation when working with very large or very small numbers. This includes numbers that are greater than 10^3 or less than 10^-3.

Q: Can I use scientific notation with negative numbers?

A: Yes, you can use scientific notation with negative numbers. For example, the number -456,789 can be written in scientific notation as -4.56789 × 10^5.

Q: Can I use scientific notation with decimal numbers?

A: Yes, you can use scientific notation with decimal numbers. For example, the number 4.56789 can be written in scientific notation as 4.56789 × 10^0.

Conclusion