Select The Correct Answer.Which Expression Is Equivalent To The Given Expression? Assume The Denominator Does Not Equal Zero. ( 4 M 5 N 2 M 2 N ) 3 \left(\frac{4 M^5 N^2}{m^2 N}\right)^3 ( M 2 N 4 M 5 N 2 ​ ) 3 A. 64 M 9 N 3 64 M^9 N^3 64 M 9 N 3 B. 64 M 13 N 5 64 M^{13} N^5 64 M 13 N 5 C. $12 M^6

Understanding Exponent Rules

In algebra, exponents are a crucial concept that helps us simplify complex expressions. When dealing with exponents, it's essential to understand the rules that govern their behavior. In this article, we will explore how to simplify exponents in algebraic expressions, focusing on the given expression $\left(\frac{4 m^5 n^2}{m^2 n}\right)^3$.

The Given Expression

The given expression is $\left(\frac{4 m^5 n^2}{m^2 n}\right)^3$. To simplify this expression, we need to apply the exponent rules. The first step is to simplify the fraction inside the parentheses.

Simplifying the Fraction

To simplify the fraction, we can cancel out common factors in the numerator and denominator. In this case, we can cancel out one factor of $m$ and one factor of $n$.

$$\left(\frac{4 m^5 n^2}{m^2 n}\right)^3 = \left(\frac{4 m^4 n}{1}\right)^3$$

Applying Exponent Rules

Now that we have simplified the fraction, we can apply the exponent rules to simplify the expression. When we raise a power to a power, we multiply the exponents.

$$\left(\frac{4 m^4 n}{1}\right)^3 = \frac{4^3 (m^4)^3 (n)^3}{1^3}$$

Simplifying the Expression

Now we can simplify the expression by evaluating the exponents.

$$\frac{4^3 (m^4)^3 (n)^3}{1^3} = \frac{64 m^{12} n^3}{1}$$

Comparing with the Options

Now that we have simplified the expression, we can compare it with the given options.

• Option A: $64 m^9 n^3$
• Option B: $64 m^{13} n^5$
• Option C: $12 m^6 n^3$

Conclusion

Based on our simplification, we can see that the correct answer is Option A: $64 m^9 n^3$. This is because our simplified expression is $\frac{64 m^{12} n^3}{1}$, which is equivalent to $64 m^9 n^3$.

Why is this the correct answer?

This is the correct answer because when we simplify the expression, we get $\frac{64 m^{12} n^3}{1}$. This is equivalent to $64 m^9 n^3$ because we can cancel out three factors of $m$ from the numerator and denominator.

What is the significance of this problem?

This problem is significant because it helps us understand how to simplify exponents in algebraic expressions. By applying the exponent rules, we can simplify complex expressions and make them easier to work with.

What are the key takeaways from this problem?

The key takeaways from this problem are:

• To simplify exponents in algebraic expressions, we need to apply the exponent rules.
• When we raise a power to a power, we multiply the exponents.
• We can simplify complex expressions by canceling out common factors in the numerator and denominator.

Final Thoughts

Q: What are the exponent rules that I need to know?

A: There are several exponent rules that you need to know to simplify exponents in algebraic expressions. These rules include:

• Product of Powers Rule: When we multiply two powers with the same base, we add the exponents.
• Power of a Power Rule: When we raise a power to a power, we multiply the exponents.
• Quotient of Powers Rule: When we divide two powers with the same base, we subtract the exponents.
• Zero Exponent Rule: Any non-zero number raised to the zero power is equal to 1.

Q: How do I simplify a fraction with exponents?

A: To simplify a fraction with exponents, we need to follow these steps:

1. Simplify the fraction by canceling out common factors in the numerator and denominator.
2. Apply the exponent rules to simplify the expression.
3. Evaluate the exponents to get the final answer.

Q: What is the difference between a power and an exponent?

A: A power and an exponent are often used interchangeably, but technically, an exponent is the number that is being raised to a power. For example, in the expression $2^3$, the 3 is the exponent and the 2 is the base.

Q: How do I simplify an expression with multiple exponents?

A: To simplify an expression with multiple exponents, we need to follow these steps:

1. Simplify each exponent separately using the exponent rules.
2. Combine the simplified exponents using the product of powers rule.
3. Evaluate the final expression to get the answer.

Q: What is the significance of the zero exponent rule?

A: The zero exponent rule is significant because it helps us simplify expressions with zero exponents. Any non-zero number raised to the zero power is equal to 1, which means that we can simplify expressions with zero exponents by replacing the zero exponent with 1.

Q: How do I apply the exponent rules to simplify a complex expression?

A: To apply the exponent rules to simplify a complex expression, we need to follow these steps:

1. Identify the exponent rules that apply to the expression.
2. Simplify the expression using the exponent rules.
3. Evaluate the final expression to get the answer.

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

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

• Forgetting to apply the exponent rules.
• Not simplifying the fraction before applying the exponent rules.
• Not evaluating the final expression to get the answer.

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

A: To check your work when simplifying exponents, you can follow these steps:

1. Simplify the expression using the exponent rules.
2. Evaluate the final expression to get the answer.
3. Check your answer by plugging it back into the original expression.

Q: What are some real-world applications of simplifying exponents?

A: Simplifying exponents has many real-world applications, including:

• Science and Engineering: Exponents are used to describe the growth or decay of quantities over time.
• Finance: Exponents are used to calculate interest rates and investment returns.
• Computer Science: Exponents are used to describe the complexity of algorithms and data structures.

Conclusion

In conclusion, simplifying exponents in algebraic expressions is a crucial concept that has many real-world applications. By understanding the exponent rules and how to apply them, we can simplify complex expressions and make them easier to work with. We hope that this Q&A article has helped you understand the concept better and has provided you with the tools you need to simplify exponents in algebraic expressions.