Which Expression Is Equivalent To $\left(m^8 N^{-4}\right)^{\frac{1}{4}}$?A. $\frac{m^2}{n}$ B. $m N$ C. $m^4 N$ D. $\frac{1}{m N}$

Understanding Exponents and Their Properties

Exponents are a fundamental concept in mathematics, used to represent repeated multiplication of a number. In this article, we will explore the properties of exponents and how to simplify expressions involving exponents. We will also apply these concepts to solve a specific problem involving exponents.

The Power of a Product Rule

The power of a product rule states that for any numbers $a$ and $b$ and any exponent $n$, $(ab)^n = a^n b^n$. This rule allows us to simplify expressions involving the product of two numbers raised to a power.

The Power of a Power Rule

The power of a power rule states that for any number $a$ and any exponents $m$ and $n$, $(a^m)^n = a^{m \cdot n}$. This rule allows us to simplify expressions involving the product of two numbers raised to a power.

Simplifying the Given Expression

The given expression is $\left(m^8 n^{-4}\right)^{\frac{1}{4}}$. To simplify this expression, we can apply the power of a product rule and the power of a power rule.

Step 1: Apply the Power of a Product Rule

Using the power of a product rule, we can rewrite the expression as:

$$\left(m^8 n^{-4}\right)^{\frac{1}{4}} = \left(m^8\right)^{\frac{1}{4}} \left(n^{-4}\right)^{\frac{1}{4}}$$

Step 2: Apply the Power of a Power Rule

Using the power of a power rule, we can simplify each term:

$$\left(m^8\right)^{\frac{1}{4}} = m^{8 \cdot \frac{1}{4}} = m^2$$

$$\left(n^{-4}\right)^{\frac{1}{4}} = n^{-4 \cdot \frac{1}{4}} = n^{-1}$$

Step 3: Simplify the Expression

Now that we have simplified each term, we can rewrite the expression as:

$$\left(m^8 n^{-4}\right)^{\frac{1}{4}} = m^2 n^{-1}$$

Which Expression is Equivalent?

The simplified expression is $m^2 n^{-1}$. To determine which of the given options is equivalent, we can rewrite each option in terms of exponents.

Option A: $\frac{m^2}{n}$

Using the rule for negative exponents, we can rewrite this option as:

$$\frac{m^2}{n} = m^2 n^{-1}$$

Option B: $m n$

This option is not equivalent to the simplified expression.

Option C: $m^4 n$

This option is not equivalent to the simplified expression.

Option D: $\frac{1}{m n}$

This option is not equivalent to the simplified expression.

Conclusion

In conclusion, the expression $\left(m^8 n^{-4}\right)^{\frac{1}{4}}$ is equivalent to $m^2 n^{-1}$, which can be rewritten as $\frac{m^2}{n}$. Therefore, the correct answer is:

A. $\frac{m^2}{n}$

Final Answer

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Q: What is the power of a product rule?

A: The power of a product rule states that for any numbers $a$ and $b$ and any exponent $n$, $(ab)^n = a^n b^n$. This rule allows us to simplify expressions involving the product of two numbers raised to a power.

Q: What is the power of a power rule?

A: The power of a power rule states that for any number $a$ and any exponents $m$ and $n$, $(a^m)^n = a^{m \cdot n}$. This rule allows us to simplify expressions involving the product of two numbers raised to a power.

Q: How do I simplify an expression involving exponents?

A: To simplify an expression involving exponents, you can apply the power of a product rule and the power of a power rule. Start by rewriting the expression using the power of a product rule, and then simplify each term using the power of a power rule.

Q: What is the rule for negative exponents?

A: The rule for negative exponents states that for any number $a$ and any exponent $n$, $a^{-n} = \frac{1}{a^n}$. This rule allows us to rewrite expressions involving negative exponents in terms of positive exponents.

Q: How do I rewrite an expression involving a negative exponent?

A: To rewrite an expression involving a negative exponent, you can apply the rule for negative exponents. For example, if you have the expression $a^{-n}$, you can rewrite it as $\frac{1}{a^n}$.

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent represents repeated multiplication of a number, while a negative exponent represents repeated division of a number. For example, $a^3$ represents $a \cdot a \cdot a$, while $a^{-3}$ represents $\frac{1}{a \cdot a \cdot a}$.

Q: How do I simplify an expression involving a product of two numbers raised to a power?

A: To simplify an expression involving a product of two numbers raised to a power, you can apply the power of a product rule. For example, if you have the expression $(ab)^n$, you can rewrite it as $a^n b^n$.

Q: What is the final answer to the problem $\left(m^8 n^{-4}\right)^{\frac{1}{4}}$?

A: The final answer to the problem $\left(m^8 n^{-4}\right)^{\frac{1}{4}}$ is $\frac{m^2}{n}$.

Q: Why is it important to simplify expressions involving exponents?

A: Simplifying expressions involving exponents is important because it allows us to rewrite complex expressions in a simpler form, making it easier to solve problems and understand mathematical concepts.

Q: What are some common mistakes to avoid when simplifying expressions involving exponents?

A: Some common mistakes to avoid when simplifying expressions involving exponents include:

• Forgetting to apply the power of a product rule
• Forgetting to apply the power of a power rule
• Not rewriting negative exponents correctly
• Not simplifying expressions fully

Conclusion

In conclusion, simplifying expressions involving exponents is an important skill to master in mathematics. By understanding the power of a product rule, the power of a power rule, and the rule for negative exponents, you can simplify complex expressions and solve problems with ease. Remember to apply these rules carefully and avoid common mistakes to ensure accurate results.