What Is The Value Of $m$ In The Quotient Of Powers $\frac{9 {-3}}{9 2}=\frac{1}{9^m}$? M = □ M = \square M = □ What Is The Value Of $ N N N [/tex] In The Quotient Of Powers $\frac{x {-4}}{x {-6}}=x^n$?$n =

Introduction

When dealing with exponents, it's essential to understand the rules that govern their behavior, especially when it comes to quotient of powers. In this article, we will delve into the world of exponents and explore the value of $m$ in the quotient of powers $\frac{9^{-3}}{92}=\frac{1}{9^m}$ and the value of $n$ in the quotient of powers $\frac{x^{-4}}{x{-6}}=x^n$.

The Quotient of Powers Rule

The quotient of powers rule states that when dividing two powers 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 and $m$ and $n$ are the exponents.

Applying the Quotient of Powers Rule to the First Equation

Let's apply the quotient of powers rule to the first equation:

$$\frac{9^{-3}}{9^2}=\frac{1}{9^m}$$

Using the quotient of powers rule, we can rewrite the equation as:

$$9^{-3-2} = 9^{-5}$$

Since the bases are the same, we can equate the exponents:

$$-3-2 = -5$$

However, we are interested in finding the value of $m$, not the value of the exponent. To do this, we can rewrite the equation as:

$$\frac{1}{9^m} = 9^{-5}$$

Since the bases are the same, we can equate the exponents:

$$-m = -5$$

Solving for $m$, we get:

$$m = 5$$

Applying the Quotient of Powers Rule to the Second Equation

Let's apply the quotient of powers rule to the second equation:

$$\frac{x^{-4}}{x^{-6}}=x^n$$

Using the quotient of powers rule, we can rewrite the equation as:

$$x^{-4-(-6)} = x^2$$

Simplifying the exponent, we get:

$$x^2 = x^2$$

Since the bases are the same, we can equate the exponents:

$$-4-(-6) = 2$$

Simplifying the equation, we get:

$$2 = 2$$

However, we are interested in finding the value of $n$, not the value of the exponent. To do this, we can rewrite the equation as:

$$x^n = x^2$$

Since the bases are the same, we can equate the exponents:

$$n = 2$$

Conclusion

In conclusion, the value of $m$ in the quotient of powers $\frac{9^{-3}}{92}=\frac{1}{9^m}$ is $5$, and the value of $n$ in the quotient of powers $\frac{x^{-4}}{x{-6}}=x^n$ is $2$. These values are obtained by applying the quotient of powers rule and equating the exponents.

Examples and Applications

Here are some examples and applications of the quotient of powers rule:

• Example 1: Simplify the expression $\frac{2^3}{25}$ using the quotient of powers rule.

  Using the quotient of powers rule, we can rewrite the expression as:

  $$2^{3-5} = 2^{-2}$$

  Simplifying the exponent, we get:

  $$2^{-2} = \frac{1}{2^2} = \frac{1}{4}$$

• Example 2: Simplify the expression $\frac{x^2}{x4}$ using the quotient of powers rule.

  Using the quotient of powers rule, we can rewrite the expression as:

  $$x^{2-4} = x^{-2}$$

  Simplifying the exponent, we get:

  $$x^{-2} = \frac{1}{x^2}$$

Tips and Tricks

Here are some tips and tricks for working with exponents:

• Tip 1: When dividing two powers with the same base, subtract the exponents.

  For example, $\frac{a^m}{an} = a^{m-n}$

• Tip 2: When simplifying an expression with exponents, try to rewrite the expression using the quotient of powers rule.

  For example, $\frac{2^3}{25} = 2^{3-5} = 2^{-2} = \frac{1}{2^2} = \frac{1}{4}$

Common Mistakes

Here are some common mistakes to avoid when working with exponents:

• Mistake 1: Not subtracting the exponents when dividing two powers with the same base.

  For example, $\frac{a^m}{an} \neq a^m + a^n$

• Mistake 2: Not simplifying the exponent when rewriting an expression using the quotient of powers rule.

  For example, $\frac{2^3}{25} \neq 2^{3+5} = 2^8$

Conclusion

In conclusion, the quotient of powers rule is a powerful tool for simplifying expressions with exponents. By understanding the rules that govern the behavior of exponents, we can simplify complex expressions and solve problems with ease. Remember to subtract the exponents when dividing two powers with the same base, and try to rewrite the expression using the quotient of powers rule when simplifying an expression with exponents.

Introduction

In our previous article, we explored the quotient of powers rule and its application in simplifying expressions with exponents. In this article, we will answer some of the most frequently asked questions about the quotient of powers rule.

Q: What is the quotient of powers rule?

A: The quotient of powers rule is a mathematical rule that states that when dividing two powers with the same base, we subtract the exponents. Mathematically, this can be represented as:

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

Q: How do I apply the quotient of powers rule?

A: To apply the quotient of powers rule, simply subtract the exponents when dividing two powers with the same base. For example:

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

Q: What if the exponents are negative?

A: If the exponents are negative, we can still apply the quotient of powers rule by subtracting the exponents. For example:

$$\frac{2^{-3}}{2^{-5}} = 2^{-3-(-5)} = 2^{2} = 4$$

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

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

Q: How do I simplify expressions with exponents using the quotient of powers rule?

A: To simplify expressions with exponents using the quotient of powers rule, try to rewrite the expression using the quotient of powers rule. For example:

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

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

A: Some common mistakes to avoid when working with exponents include:

• Not subtracting the exponents when dividing two powers with the same base.
• Not simplifying the exponent when rewriting an expression using the quotient of powers rule.

Q: How do I check my work when simplifying expressions with exponents?

A: To check your work when simplifying expressions with exponents, try to rewrite the expression using the quotient of powers rule and simplify the exponent. For example:

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

Conclusion

In conclusion, the quotient of powers rule is a powerful tool for simplifying expressions with exponents. By understanding the rules that govern the behavior of exponents, we can simplify complex expressions and solve problems with ease. Remember to subtract the exponents when dividing two powers with the same base, and try to rewrite the expression using the quotient of powers rule when simplifying an expression with exponents.

Additional Resources

For more information on the quotient of powers rule and its application in simplifying expressions with exponents, check out the following resources:

• Mathway: A online math problem solver that can help you simplify expressions with exponents.
• Khan Academy: A free online learning platform that offers video lessons and practice exercises on exponents and other math topics.
• Math Open Reference: A free online math reference book that offers detailed explanations and examples of exponents and other math topics.

Practice Exercises

Try the following practice exercises to test your understanding of the quotient of powers rule:

• Exercise 1: Simplify the expression $\frac{3^4}{36}$ using the quotient of powers rule.
• Exercise 2: Simplify the expression $\frac{x^2}{x4}$ using the quotient of powers rule.
• Exercise 3: Simplify the expression $\frac{2^{-3}}{2{-5}}$ using the quotient of powers rule.

Answer Key

Here are the answers to the practice exercises:

• Exercise 1: $3^{4-6} = 3^{-2} = \frac{1}{3^2} = \frac{1}{9}$
• Exercise 2: $x^{2-4} = x^{-2} = \frac{1}{x^2}$
• Exercise 3: $2^{-3-(-5)} = 2^{2} = 4$