Which Sign Makes The Statement True?$\[ 7.1 \times 10^{14} \, ? \, 7.31 \times 10^{14} \\]A. $\[ \ \textgreater \ \\]B. $\[ \ \textless \ \\]C. $\[ = \\]

Feb 27, 2025 by ADMIN 158 views

Which Sign Makes the Statement True?

Understanding the Problem

In this problem, we are given two numbers in scientific notation and asked to determine which sign makes the statement true. The numbers are $7.1 \times 10^{14}$ and $7.31 \times 10^{14}$ . We need to compare these two numbers and decide whether the first number is greater than, less than, or equal to the second number.

Scientific Notation

Before we proceed, let's briefly review scientific notation. Scientific notation is a way of expressing very large or very small numbers in a more manageable form. It consists of a number between 1 and 10 multiplied by a power of 10. For example, the number 456,000 can be written in scientific notation as $4.56 \times 10^{5}$ .

Comparing Numbers in Scientific Notation

To compare numbers in scientific notation, we need to compare the numbers themselves and the powers of 10. If the powers of 10 are the same, we can simply compare the numbers. If the powers of 10 are different, we need to adjust the numbers to have the same power of 10 before comparing them.

Comparing the Numbers

Let's compare the two numbers given in the problem: $7.1 \times 10^{14}$ and $7.31 \times 10^{14}$ . Since the powers of 10 are the same, we can simply compare the numbers themselves. The number $7.1$ is less than the number $7.31$ .

Conclusion

Based on the comparison, we can conclude that the first number, $7.1 \times 10^{14}$ , is less than the second number, $7.31 \times 10^{14}$ . Therefore, the correct answer is:

B. ${ \ \textless \ }$

Why is this the Correct Answer?

This is the correct answer because the number $7.1$ is less than the number $7.31$ . When we compare numbers in scientific notation, we need to compare the numbers themselves and the powers of 10. In this case, the powers of 10 are the same, so we can simply compare the numbers. Since $7.1$ is less than $7.31$ , the first number is less than the second number.

What if the Powers of 10 are Different?

If the powers of 10 are different, we need to adjust the numbers to have the same power of 10 before comparing them. For example, if we have the numbers $4.56 \times 10^{5}$ and $3.21 \times 10^{6}$ , we need to adjust the numbers to have the same power of 10. We can do this by dividing the first number by $10^{5}$ and the second number by $10^{6}$ . This gives us $4.56$ and $3.21$ , respectively. Since $4.56$ is greater than $3.21$ , the first number is greater than the second number.

Conclusion

In conclusion, to determine which sign makes the statement true, we need to compare the numbers in scientific notation. If the powers of 10 are the same, we can simply compare the numbers. If the powers of 10 are different, we need to adjust the numbers to have the same power of 10 before comparing them. In this case, the first number, $7.1 \times 10^{14}$ , is less than the second number, $7.31 \times 10^{14}$ . Therefore, the correct answer is:

B. ${ \ \textless \ }$

Key Takeaways

To compare numbers in scientific notation, we need to compare the numbers themselves and the powers of 10.
If the powers of 10 are the same, we can simply compare the numbers.
If the powers of 10 are different, we need to adjust the numbers to have the same power of 10 before comparing them.
The correct answer is B. ${ \ \textless \ }$ because the number $7.1$ is less than the number $7.31$ .
Which Sign Makes the Statement True? - Q&A

Frequently Asked Questions

In this article, we will answer some frequently asked questions related to the problem of determining which sign makes the statement true.

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 number between 1 and 10 multiplied by a power of 10.

Q: How do I compare numbers in scientific notation?

A: To compare numbers in scientific notation, you need to compare the numbers themselves and the powers of 10. If the powers of 10 are the same, you can simply compare the numbers. If the powers of 10 are different, you need to adjust the numbers to have the same power of 10 before comparing them.

Q: What if the powers of 10 are different?

A: If the powers of 10 are different, you need to adjust the numbers to have the same power of 10 before comparing them. You can do this by dividing the first number by 10 raised to the power of the smaller exponent and the second number by 10 raised to the power of the larger exponent.

Q: How do I adjust the numbers to have the same power of 10?

A: To adjust the numbers to have the same power of 10, you need to divide the first number by 10 raised to the power of the difference between the two exponents. For example, if you have the numbers $4.56 \times 10^{5}$ and $3.21 \times 10^{6}$ , you need to divide the first number by $10^{5}$ and the second number by $10^{6}$ .

Q: What if I have a negative exponent?

A: If you have a negative exponent, you need to take the reciprocal of the number and change the sign of the exponent. For example, if you have the number $4.56 \times 10^{-5}$ , you need to take the reciprocal of the number and change the sign of the exponent to get $4.56 \times 10^{5}$ .

Q: Can I use a calculator to compare numbers in scientific notation?

A: Yes, you can use a calculator to compare numbers in scientific notation. Most calculators have a scientific notation mode that allows you to enter numbers in scientific notation and perform calculations.

Q: What are some common mistakes to avoid when comparing numbers in scientific notation?

A: Some common mistakes to avoid when comparing numbers in scientific notation include:

Not comparing the powers of 10
Not adjusting the numbers to have the same power of 10
Not taking into account negative exponents
Not using a calculator to check your calculations

Conclusion

In conclusion, comparing numbers in scientific notation requires attention to detail and a clear understanding of the rules for comparing numbers in scientific notation. By following the steps outlined in this article, you can confidently compare numbers in scientific notation and determine which sign makes the statement true.

Key Takeaways

To compare numbers in scientific notation, you need to compare the numbers themselves and the powers of 10.
If the powers of 10 are the same, you can simply compare the numbers.
If the powers of 10 are different, you need to adjust the numbers to have the same power of 10 before comparing them.
You can use a calculator to compare numbers in scientific notation.
Some common mistakes to avoid when comparing numbers in scientific notation include not comparing the powers of 10, not adjusting the numbers to have the same power of 10, not taking into account negative exponents, and not using a calculator to check your calculations.