Which Expression Is Equivalent To ( 4 M N M − 2 N 6 ) − 2 \left(\frac{4 M N}{m^{-2} N^6}\right)^{-2} ( M − 2 N 6 4 Mn ​ ) − 2 ? Assume M ≠ 0 , N ≠ 0 M \neq 0, N \neq 0 M  = 0 , N  = 0 .A. N 6 16 M 8 \frac{n^6}{16 M^8} 16 M 8 N 6 ​ B. N 10 16 M 6 \frac{n^{10}}{16 M^6} 16 M 6 N 10 ​ C. N 10 8 M 8 \frac{n^{10}}{8 M^8} 8 M 8 N 10 ​ D. $\frac{4

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Introduction

When dealing with exponential expressions, it's essential to understand the rules of exponents and how to simplify them. In this article, we'll focus on simplifying the expression (4mnm−2n6)−2\left(\frac{4 m n}{m^{-2} n^6}\right)^{-2} and explore the different options provided.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, a3a^3 can be written as a⋅a⋅aa \cdot a \cdot a. When dealing with exponents, there are several rules to keep in mind:

  • Product of Powers Rule: When multiplying two powers with the same base, add the exponents. For example, a2⋅a3=a2+3=a5a^2 \cdot a^3 = a^{2+3} = a^5.
  • Power of a Power Rule: When raising a power to another power, multiply the exponents. For example, (a2)3=a2⋅3=a6(a^2)^3 = a^{2 \cdot 3} = a^6.
  • Quotient of Powers Rule: When dividing two powers with the same base, subtract the exponents. For example, a2a3=a2−3=a−1\frac{a^2}{a^3} = a^{2-3} = a^{-1}.

Simplifying the Expression

To simplify the expression (4mnm−2n6)−2\left(\frac{4 m n}{m^{-2} n^6}\right)^{-2}, we'll start by applying the quotient of powers rule to the fraction inside the parentheses.

(4mnm−2n6)−2=(4mn1m2n6)−2\left(\frac{4 m n}{m^{-2} n^6}\right)^{-2} = \left(\frac{4 m n}{\frac{1}{m^2} n^6}\right)^{-2}

Next, we'll simplify the fraction by multiplying the numerator by the reciprocal of the denominator.

(4mn1m2n6)−2=(4mn⋅m2n−6)−2\left(\frac{4 m n}{\frac{1}{m^2} n^6}\right)^{-2} = \left(4 m n \cdot m^2 n^{-6}\right)^{-2}

Now, we'll apply the product of powers rule to simplify the expression inside the parentheses.

(4mn⋅m2n−6)−2=(4m1+2n1−6)−2\left(4 m n \cdot m^2 n^{-6}\right)^{-2} = \left(4 m^{1+2} n^{1-6}\right)^{-2}

(4m1+2n1−6)−2=(4m3n−5)−2\left(4 m^{1+2} n^{1-6}\right)^{-2} = \left(4 m^3 n^{-5}\right)^{-2}

Applying the Power of a Power Rule

Now that we have the expression inside the parentheses simplified, we can apply the power of a power rule to simplify the entire expression.

(4m3n−5)−2=4−2(m3)−2(n−5)−2\left(4 m^3 n^{-5}\right)^{-2} = 4^{-2} (m^3)^{-2} (n^{-5})^{-2}

4−2(m3)−2(n−5)−2=142⋅1(m3)2⋅1(n−5)24^{-2} (m^3)^{-2} (n^{-5})^{-2} = \frac{1}{4^2} \cdot \frac{1}{(m^3)^2} \cdot \frac{1}{(n^{-5})^2}

Simplifying the Expression Further

Now that we have the expression simplified, we can simplify it further by applying the quotient of powers rule.

142⋅1(m3)2⋅1(n−5)2=116⋅1m3⋅2⋅1n−5⋅2\frac{1}{4^2} \cdot \frac{1}{(m^3)^2} \cdot \frac{1}{(n^{-5})^2} = \frac{1}{16} \cdot \frac{1}{m^{3 \cdot 2}} \cdot \frac{1}{n^{-5 \cdot 2}}

116⋅1m3⋅2⋅1n−5⋅2=116⋅1m6⋅1n−10\frac{1}{16} \cdot \frac{1}{m^{3 \cdot 2}} \cdot \frac{1}{n^{-5 \cdot 2}} = \frac{1}{16} \cdot \frac{1}{m^6} \cdot \frac{1}{n^{-10}}

Final Simplification

Now that we have the expression simplified, we can simplify it further by applying the product of powers rule.

116⋅1m6⋅1n−10=116⋅n10m6\frac{1}{16} \cdot \frac{1}{m^6} \cdot \frac{1}{n^{-10}} = \frac{1}{16} \cdot \frac{n^{10}}{m^6}

116⋅n10m6=n1016m6\frac{1}{16} \cdot \frac{n^{10}}{m^6} = \frac{n^{10}}{16 m^6}

Conclusion

In conclusion, the expression (4mnm−2n6)−2\left(\frac{4 m n}{m^{-2} n^6}\right)^{-2} is equivalent to n1016m6\frac{n^{10}}{16 m^6}. This is the correct answer among the options provided.

Discussion

The expression (4mnm−2n6)−2\left(\frac{4 m n}{m^{-2} n^6}\right)^{-2} is a complex expression that requires careful simplification. By applying the rules of exponents, we can simplify the expression and arrive at the correct answer.

Final Answer

The final answer is n1016m6\boxed{\frac{n^{10}}{16 m^6}}.

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Q: What is the rule for simplifying exponential expressions?

A: The rule for simplifying exponential expressions is to apply the quotient of powers rule, product of powers rule, and power of a power rule in the correct order.

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

A: To apply the quotient of powers rule, you need to subtract the exponents when dividing two powers with the same base.

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

A: To apply the product of powers rule, you need to add the exponents when multiplying two powers with the same base.

Q: How do I apply the power of a power rule?

A: To apply the power of a power rule, you need to multiply the exponents when raising a power to another power.

Q: What is the correct order for simplifying exponential expressions?

A: The correct order for simplifying exponential expressions is to:

  1. Apply the quotient of powers rule to simplify the fraction inside the parentheses.
  2. Apply the product of powers rule to simplify the expression inside the parentheses.
  3. Apply the power of a power rule to simplify the entire expression.

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

A: To simplify an expression with negative exponents, you need to apply the rule for negative exponents, which states that a−n=1ana^{-n} = \frac{1}{a^n}.

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

A: To simplify an expression with fractional exponents, you need to apply the rule for fractional exponents, which states that amn=amna^{\frac{m}{n}} = \sqrt[n]{a^m}.

Q: What is the final answer to the expression (4mnm−2n6)−2\left(\frac{4 m n}{m^{-2} n^6}\right)^{-2}?

A: The final answer to the expression (4mnm−2n6)−2\left(\frac{4 m n}{m^{-2} n^6}\right)^{-2} is n1016m6\frac{n^{10}}{16 m^6}.

Q: Why is it essential to simplify exponential expressions?

A: It is essential to simplify exponential expressions because it helps to:

  • Reduce the complexity of the expression
  • Make it easier to evaluate
  • Identify patterns and relationships between variables

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

A: To evaluate an expression with multiple exponents, you need to apply the rules of exponents in the correct order, starting with the innermost exponent and working your way outwards.

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

A: Some common mistakes to avoid when simplifying exponential expressions include:

  • Failing to apply the rules of exponents in the correct order
  • Not simplifying negative exponents correctly
  • Not simplifying fractional exponents correctly

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

A: To check your work when simplifying exponential expressions, you need to:

  • Verify that you have applied the rules of exponents correctly
  • Check that your final answer is in the simplest form possible
  • Evaluate your final answer to ensure it is correct

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

A: Some real-world applications of simplifying exponential expressions include:

  • Calculating population growth rates
  • Modeling financial investments
  • Analyzing data in science and engineering

Q: How do I use technology to simplify exponential expressions?

A: You can use technology such as calculators or computer software to simplify exponential expressions. These tools can help you to:

  • Evaluate expressions quickly and accurately
  • Identify patterns and relationships between variables
  • Visualize complex expressions in a simplified form