What Is The Quotient Of The Following Expression? 5 − 6 5 3 \frac{5^{-6}}{5^3} 5 3 5 − 6 ​ A. 1 5 9 \frac{1}{5^9} 5 9 1 ​ B. 1 5 2 \frac{1}{5^2} 5 2 1 ​ C. 5 2 5^2 5 2 D. 5 9 5^9 5 9

Understanding Exponents and Quotient Rules

In mathematics, exponents are a shorthand way of representing repeated multiplication of a number. For example, $5^3$ means $5 \times 5 \times 5$. When dealing with exponents, there are certain rules that help us simplify expressions and find their values. One of these rules is the quotient rule, which states that when we divide two numbers with the same base, we subtract the exponents.

The Quotient Rule

The quotient rule is a fundamental concept in mathematics that helps us simplify expressions involving exponents. It states that when we divide two numbers with the same base, we subtract the exponents. Mathematically, this can be represented as:

$$\frac{a^m}{a^n} = a^{m-n}$$

where $a$ is the base, $m$ and $n$ are the exponents, and $a^{m-n}$ is the result of the division.

Applying the Quotient Rule to the Given Expression

Now, let's apply the quotient rule to the given expression: $\frac{5^{-6}}{5^3}$. Using the quotient rule, we can simplify this expression by subtracting the exponents:

$$\frac{5^{-6}}{5^3} = 5^{-6-3} = 5^{-9}$$

Simplifying the Result

Now that we have simplified the expression using the quotient rule, we can further simplify the result by expressing it in a more familiar form. Since $5^{-9}$ is equivalent to $\frac{1}{5^9}$, we can write the final result as:

$$\frac{5^{-6}}{5^3} = \frac{1}{5^9}$$

Conclusion

In conclusion, the quotient of the given expression $\frac{5^{-6}}{5^3}$ is $\frac{1}{5^9}$. This result was obtained by applying the quotient rule, which states that when we divide two numbers with the same base, we subtract the exponents. By simplifying the expression using this rule, we arrived at the final result.

Why is the Quotient Rule Important?

The quotient rule is an essential concept in mathematics that helps us simplify expressions involving exponents. It is used extensively in various mathematical operations, including division, multiplication, and exponentiation. By understanding and applying the quotient rule, we can simplify complex expressions and arrive at their values.

Real-World Applications of the Quotient Rule

The quotient rule has numerous real-world applications in various fields, including science, engineering, and finance. For example, in physics, the quotient rule is used to calculate the acceleration of an object when its velocity and time are known. In finance, the quotient rule is used to calculate the interest rate on a loan when the principal amount, time, and interest rate are known.

Common Mistakes to Avoid

When applying the quotient rule, there are several common mistakes to avoid. These include:

• Incorrectly applying the quotient rule: Make sure to subtract the exponents when dividing two numbers with the same base.
• Not simplifying the result: Take the time to simplify the result by expressing it in a more familiar form.
• Not checking the signs of the exponents: Make sure to check the signs of the exponents before applying the quotient rule.

Practice Problems

To reinforce your understanding of the quotient rule, try solving the following practice problems:

1. $\frac{2^4}{2^2} = \boxed{2^2}$
2. $\frac{3^{-2}}{3^4} = \boxed{3^{-6}}$
3. $\frac{4^3}{4^5} = \boxed{4^{-2}}$

Conclusion

Frequently Asked Questions About the Quotient Rule

The quotient rule is a fundamental concept in mathematics that helps us simplify expressions involving exponents. However, it can be a bit tricky to understand and apply, especially for beginners. In this article, we'll answer some frequently asked questions about the quotient rule to help you better understand and apply it.

Q: What is the quotient rule?

A: The quotient rule is a mathematical rule that helps us simplify expressions involving exponents. It states that when we divide two numbers with the same base, we subtract the exponents.

Q: How do I apply the quotient rule?

A: To apply the quotient rule, simply subtract the exponents of the two numbers with the same base. For example, if we have $\frac{a^m}{a^n}$, we can simplify it by subtracting the exponents: $a^{m-n}$.

Q: What if the exponents are negative?

A: If the exponents are negative, we can still apply the quotient rule by subtracting the exponents. For example, if we have $\frac{a^{-m}}{a^n}$, we can simplify it by subtracting the exponents: $a^{n-m}$.

Q: Can I apply the quotient rule to expressions with different bases?

A: No, the quotient rule only applies to expressions with the same base. If the bases are different, we cannot apply the quotient rule.

Q: How do I simplify expressions involving the quotient rule?

A: To simplify expressions involving the quotient rule, simply apply the rule and simplify the result. For example, if we have $\frac{a^m}{a^n}$, we can simplify it by subtracting the exponents: $a^{m-n}$.

Q: What are some common mistakes to avoid when applying the quotient rule?

A: Some common mistakes to avoid when applying the quotient rule include:

• Incorrectly applying the quotient rule: Make sure to subtract the exponents when dividing two numbers with the same base.
• Not simplifying the result: Take the time to simplify the result by expressing it in a more familiar form.
• Not checking the signs of the exponents: Make sure to check the signs of the exponents before applying the quotient rule.

Q: How can I practice applying the quotient rule?

A: You can practice applying the quotient rule by solving practice problems. Try solving the following practice problems:

1. $\frac{2^4}{2^2} = \boxed{2^2}$
2. $\frac{3^{-2}}{3^4} = \boxed{3^{-6}}$
3. $\frac{4^3}{4^5} = \boxed{4^{-2}}$

Q: What are some real-world applications of the quotient rule?

A: The quotient rule has numerous real-world applications in various fields, including science, engineering, and finance. For example, in physics, the quotient rule is used to calculate the acceleration of an object when its velocity and time are known. In finance, the quotient rule is used to calculate the interest rate on a loan when the principal amount, time, and interest rate are known.

Conclusion

In conclusion, the quotient rule is a fundamental concept in mathematics that helps us simplify expressions involving exponents. By understanding and applying the quotient rule, we can simplify complex expressions and arrive at their values. Remember to avoid common mistakes, such as incorrectly applying the quotient rule and not simplifying the result. With practice and patience, you will become proficient in applying the quotient rule and solving complex mathematical problems.