The Quotient of Two Numbers in Scientific Notation

Oct 13, 2025 by windailys.com 6 views

Understanding the Problem

When dealing with numbers in scientific notation, it's essential to understand the rules of operations and how to manipulate them. In this problem, we are given two numbers in scientific notation: $$8.4 \times 10^9$$ and $$5.6 \times 10^{27}$$. We need to find the value of a number $$n$$ such that the quotient of $$8.4 \times 10^9$$ and $$n$$ results in $$5.6 \times 10^{27}$$.

The Quotient of Two Numbers in Scientific Notation

To find the value of $$n$$, we can use the rule for dividing numbers in scientific notation. When dividing two numbers in scientific notation, we divide the coefficients (the numbers in front of the powers of 10) and subtract the exponents of the powers of 10.

The Formula for Dividing Numbers in Scientific Notation

The formula for dividing numbers in scientific notation is:

\frac{a \times 10^m}{b \times 10^n} = \frac{a}{b} \times 10^{m-n}

where $$a$$ and $$b$$ are the coefficients, and $$m$$ and $$n$$ are the exponents.

Applying the Formula to the Problem

Using the formula, we can rewrite the quotient of $$8.4 \times 10^9$$ and $$n$$ as:

\frac{8.4 \times 10^9}{n} = 5.6 \times 10^{27}

Simplifying the Equation

We can simplify the equation by dividing the coefficients and subtracting the exponents:

\frac{8.4}{n} = 5.6 \times 10^{27-9}

\frac{8.4}{n} = 5.6 \times 10^{18}

Solving for $$n$$

To solve for $$n$$, we can multiply both sides of the equation by $$n$$:

8.4 = 5.6 \times 10^{18} \times n

Dividing Both Sides by $$5.6 \times 10^{18}$$

We can divide both sides of the equation by $$5.6 \times 10^{18}$$ to isolate $$n$$:

n = \frac{8.4}{5.6 \times 10^{18}}

Simplifying the Expression

We can simplify the expression by dividing the coefficients and subtracting the exponent:

n = \frac{8.4}{5.6} \times 10^{-18}

n = 1.5 \times 10^{-18}

Conclusion

The value of $$n$$ is $$1.5 \times 10^{-18}$$.

Discussion

This problem requires a good understanding of the rules of operations for numbers in scientific notation. The key concept is to use the formula for dividing numbers in scientific notation and to simplify the equation by dividing the coefficients and subtracting the exponents.

Common Mistakes

One common mistake is to forget to subtract the exponents when dividing numbers in scientific notation. Another mistake is to not simplify the equation by dividing the coefficients.

Real-World Applications

This problem has real-world applications in various fields, such as physics, engineering, and chemistry, where numbers in scientific notation are commonly used to represent large or small quantities.

Practice Problems

What is the quotient of $$2.5 \times 10^6$$ and $$3.2 \times 10^{12}$$?
What is the value of $$n$$ such that the quotient of $$4.8 \times 10^8$$ and $$n$$ results in $$2.1 \times 10^{25}$$?

Solutions

$\frac{2.5 \times 10^6}{3.2 \times 10^{12}} = 7.8 \times 10^{-7}$
$n = 1.2 \times 10^{-17}$$<br/>$

Frequently Asked Questions

Q: What is the quotient of two numbers in scientific notation?

A: The quotient of two numbers in scientific notation is the result of dividing one number by another, where both numbers are expressed in scientific notation.

Q: How do I divide numbers in scientific notation?

A: To divide numbers in scientific notation, you divide the coefficients (the numbers in front of the powers of 10) and subtract the exponents of the powers of 10.

Q: What is the formula for dividing numbers in scientific notation?

A: The formula for dividing numbers in scientific notation is:

\frac{a \times 10^m}{b \times 10^n} = \frac{a}{b} \times 10^{m-n}

where $$a$$ and $$b$$ are the coefficients, and $$m$$ and $$n$$ are the exponents.

Q: How do I simplify the equation after dividing numbers in scientific notation?

A: To simplify the equation, you divide the coefficients and subtract the exponents.

Q: What is the value of $$n$$ such that the quotient of $$8.4 \times 10^9$$ and $$n$$ results in $$5.6 \times 10^{27}$$?

A: The value of $$n$$ is $$1.5 \times 10^{-18}$$.

Q: What is the quotient of $$2.5 \times 10^6$$ and $$3.2 \times 10^{12}$$?

A: The quotient of $$2.5 \times 10^6$$ and $$3.2 \times 10^{12}$$ is $$7.8 \times 10^{-7}$$.

Q: What is the value of $$n$$ such that the quotient of $$4.8 \times 10^8$$ and $$n$$ results in $$2.1 \times 10^{25}$$?

A: The value of $$n$$ is $$1.2 \times 10^{-17}$$.

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

A: Some common mistakes to avoid when dividing numbers in scientific notation include forgetting to subtract the exponents and not simplifying the equation by dividing the coefficients.

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

A: Dividing numbers in scientific notation has real-world applications in various fields, such as physics, engineering, and chemistry, where numbers in scientific notation are commonly used to represent large or small quantities.

Q: How can I practice dividing numbers in scientific notation?

A: You can practice dividing numbers in scientific notation by working through practice problems, such as the ones listed below.

Practice Problems

What is the quotient of $$1.2 \times 10^5$$ and $$2.5 \times 10^{11}$$?
What is the value of $$n$$ such that the quotient of $$3.4 \times 10^7$$ and $$n$$ results in $$1.8 \times 10^{30}$$?
What is the quotient of $$4.9 \times 10^3$$ and $$6.2 \times 10^{15}$$?

Solutions

$\frac{1.2 \times 10^5}{2.5 \times 10^{11}} = 4.8 \times 10^{-7}$
$n = 1.9 \times 10^{-23}$
$\frac{4.9 \times 10^3}{6.2 \times 10^{15}} = 7.9 \times 10^{-13}$