Study The Expression. What Is The Value Of The Expression When $\left(4 M^ -3} N 2\right) 2$ Is Evaluated?1. Apply The Power Of A Power ${ 4^2 M^{-6 N^4 }$2. Substitute Values For Variables: $[ (4)^2 (2)^{-6}

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Understanding the Problem

When evaluating mathematical expressions, it's essential to follow the correct order of operations and apply the relevant rules to simplify the expression. In this case, we're given the expression $\left(4 m^{-3} n2\right)2$ and asked to find its value. To do this, we'll apply the power of a power rule and then substitute values for the variables.

Applying the Power of a Power Rule

The power of a power rule states that when a power is raised to another power, we multiply the exponents. In this case, we have $\left(4 m^{-3} n2\right)2$. To apply the power of a power rule, we'll raise each component of the expression to the power of 2.

(4m−3n2)2=42(m−3)2(n2)2\left(4 m^{-3} n^2\right)^2 = 4^2 \left(m^{-3}\right)^2 \left(n^2\right)^2

Simplifying the Expression

Now that we've applied the power of a power rule, we can simplify the expression by evaluating the exponents.

42=164^2 = 16

(m−3)2=m−6\left(m^{-3}\right)^2 = m^{-6}

(n2)2=n4\left(n^2\right)^2 = n^4

Combining the Simplified Components

Now that we've simplified each component of the expression, we can combine them to get the final result.

16m−6n416 m^{-6} n^4

Substituting Values for Variables

Now that we have the simplified expression, we can substitute values for the variables to find the final value. However, in this case, we're not given specific values for the variables. Instead, we're asked to evaluate the expression in terms of the variables.

Evaluating the Expression in Terms of Variables

Since we're not given specific values for the variables, we can leave the expression in terms of the variables. The final value of the expression is $16 m^{-6} n^4$.

Conclusion

In this article, we've studied the expression $\left(4 m^{-3} n2\right)2$ and evaluated its value using the power of a power rule. We've simplified the expression and combined the simplified components to get the final result. We've also discussed the importance of following the correct order of operations and applying the relevant rules to simplify mathematical expressions.

Frequently Asked Questions

  • What is the power of a power rule? The power of a power rule states that when a power is raised to another power, we multiply the exponents.
  • How do we apply the power of a power rule? To apply the power of a power rule, we raise each component of the expression to the power of the outer exponent.
  • What is the final value of the expression $\left(4 m^{-3} n2\right)2$? The final value of the expression is $16 m^{-6} n^4$.

Final Answer

The final answer is $16 m^{-6} n^4$.

Frequently Asked Questions

In this article, we'll answer some of the most frequently asked questions related to the expression $\left(4 m^{-3} n2\right)2$ and its evaluation.

Q: What is the power of a power rule?

A: The power of a power rule states that when a power is raised to another power, we multiply the exponents. This rule is essential in simplifying complex mathematical expressions.

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

A: To apply the power of a power rule, we raise each component of the expression to the power of the outer exponent. This means that we multiply the exponents of each component.

Q: What is the final value of the expression $\left(4 m^{-3} n2\right)2$?

A: The final value of the expression is $16 m^{-6} n^4$.

Q: Why is it essential to follow the correct order of operations when evaluating mathematical expressions?

A: Following the correct order of operations is crucial when evaluating mathematical expressions. It ensures that we perform the operations in the correct order and avoid errors.

Q: Can we substitute values for the variables in the expression $\left(4 m^{-3} n2\right)2$?

A: Yes, we can substitute values for the variables in the expression. However, in this case, we're not given specific values for the variables, so we leave the expression in terms of the variables.

Q: What is the significance of the exponent $m^{-6}$ in the expression $16 m^{-6} n^4$?

A: The exponent $m^{-6}$ indicates that the variable $m$ is raised to the power of $-6$. This means that the value of $m$ will be divided by $m^6$.

Q: Can we simplify the expression $16 m^{-6} n^4$ further?

A: Yes, we can simplify the expression further by combining the terms. However, in this case, the expression is already simplified, and we can leave it as is.

Q: What is the final answer to the expression $\left(4 m^{-3} n2\right)2$?

A: The final answer to the expression is $16 m^{-6} n^4$.

Additional Resources

If you're looking for additional resources to help you understand the expression $\left(4 m^{-3} n2\right)2$ and its evaluation, here are some suggestions:

  • Math textbooks: Check out math textbooks that cover algebra and exponent rules.
  • Online resources: Visit online resources such as Khan Academy, Mathway, or Wolfram Alpha to learn more about exponent rules and expression evaluation.
  • Practice problems: Practice evaluating expressions with different exponents and variables to improve your skills.

Conclusion

In this article, we've answered some of the most frequently asked questions related to the expression $\left(4 m^{-3} n2\right)2$ and its evaluation. We've covered topics such as the power of a power rule, expression simplification, and variable substitution. If you have any further questions or need additional resources, feel free to ask.

Final Answer

The final answer is $16 m^{-6} n^4$.