Simplify The Expression. Write The Answer With Positive Exponents Only. Assume All Variables Represent Nonzero Real Numbers.$\[ \left(3 M^5 N^{-7}\right)^{-3} = \square \\]

Understanding the Problem

In this problem, we are given an expression with a variable base and an exponent. The expression is $\left(3 m^5 n^{-7}\right)^{-3}$, and we need to simplify it by writing the answer with positive exponents only. We are also given that all variables represent nonzero real numbers.

Recall the Power Rule of Exponents

To simplify the expression, we need to recall the power rule of exponents, which states that for any nonzero real numbers $a$ and $b$ and any integers $m$ and $n$, we have:

$$\left(a^m\right)^n = a^{m \cdot n}$$

Apply the Power Rule to the Expression

Using the power rule, we can rewrite the expression as:

$$\left(3 m^5 n^{-7}\right)^{-3} = 3^{-3} \cdot \left(m^5\right)^{-3} \cdot \left(n^{-7}\right)^{-3}$$

Simplify the Expression Further

Now, we can simplify the expression further by applying the power rule again:

$$3^{-3} \cdot \left(m^5\right)^{-3} \cdot \left(n^{-7}\right)^{-3} = 3^{-3} \cdot m^{-15} \cdot n^{21}$$

Write the Answer with Positive Exponents Only

To write the answer with positive exponents only, we need to get rid of the negative exponents. We can do this by taking the reciprocal of the base and changing the sign of the exponent:

$$3^{-3} \cdot m^{-15} \cdot n^{21} = \frac{1}{3^3} \cdot \frac{1}{m^{15}} \cdot n^{21}$$

Simplify the Expression to Get the Final Answer

Finally, we can simplify the expression to get the final answer:

$$\frac{1}{3^3} \cdot \frac{1}{m^{15}} \cdot n^{21} = \frac{n^{21}}{3^3 m^{15}}$$

Conclusion

In this problem, we simplified the expression $\left(3 m^5 n^{-7}\right)^{-3}$ by writing the answer with positive exponents only. We used the power rule of exponents to simplify the expression and then took the reciprocal of the base to get rid of the negative exponents. The final answer is $\frac{n^{21}}{3^3 m^{15}}$.

Key Takeaways

• The power rule of exponents states that for any nonzero real numbers $a$ and $b$ and any integers $m$ and $n$, we have $\left(a^m\right)^n = a^{m \cdot n}$.
• To simplify an expression with a variable base and an exponent, we can use the power rule to rewrite the expression and then simplify it further.
• To write an expression with negative exponents, we can take the reciprocal of the base and change the sign of the exponent.

Practice Problems

• Simplify the expression $\left(2 x^3 y^{-4}\right)^{-2}$ by writing the answer with positive exponents only.
• Simplify the expression $\left(5 a^2 b^{-3}\right)^{-4}$ by writing the answer with positive exponents only.

Glossary of Terms

• Power rule of exponents: A rule that states that for any nonzero real numbers $a$ and $b$ and any integers $m$ and $n$, we have $\left(a^m\right)^n = a^{m \cdot n}$.
• Negative exponent: An exponent that is less than zero, such as $-3$ or $-4$.
• Positive exponent: An exponent that is greater than zero, such as $3$ or $4$.
• Reciprocal: The inverse of a number, such as $\frac{1}{a}$ or $\frac{1}{b}$.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton
  Simplify the Expression with Positive Exponents: Q&A =====================================================

Q: What is the power rule of exponents?

A: The power rule of exponents states that for any nonzero real numbers $a$ and $b$ and any integers $m$ and $n$, we have $\left(a^m\right)^n = a^{m \cdot n}$.

Q: How do I simplify an expression with a variable base and an exponent?

A: To simplify an expression with a variable base and an exponent, you can use the power rule to rewrite the expression and then simplify it further.

Q: What is a negative exponent?

A: A negative exponent is an exponent that is less than zero, such as $-3$ or $-4$.

Q: How do I get rid of a negative exponent?

A: To get rid of a negative exponent, you can take the reciprocal of the base and change the sign of the exponent.

Q: What is the reciprocal of a number?

A: The reciprocal of a number is the inverse of the number, such as $\frac{1}{a}$ or $\frac{1}{b}$.

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

A: To simplify an expression with multiple variables and exponents, you can use the power rule to rewrite the expression and then simplify it further.

Q: What is the final answer to the expression $\left(3 m^5 n^{-7}\right)^{-3}$?

A: The final answer to the expression $\left(3 m^5 n^{-7}\right)^{-3}$ is $\frac{n^{21}}{3^3 m^{15}}$.

Q: Can you give me some practice problems to try?

A: Yes, here are some practice problems to try:

• Simplify the expression $\left(2 x^3 y^{-4}\right)^{-2}$ by writing the answer with positive exponents only.
• Simplify the expression $\left(5 a^2 b^{-3}\right)^{-4}$ by writing the answer with positive exponents only.

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

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

• Not using the power rule correctly
• Not getting rid of negative exponents
• Not simplifying the expression further

Q: How do I know if I have simplified an expression correctly?

A: To know if you have simplified an expression correctly, you can check your work by plugging in some values for the variables and exponents. If the expression simplifies to the same value, then you have simplified it correctly.

Q: Can you give me some tips for simplifying expressions with exponents?

A: Yes, here are some tips for simplifying expressions with exponents:

• Use the power rule correctly
• Get rid of negative exponents
• Simplify the expression further
• Check your work by plugging in some values for the variables and exponents

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

A: Some real-world applications of simplifying expressions with exponents include:

• Calculating interest rates and investments
• Solving problems in physics and engineering
• Working with computer algorithms and programming

Q: Can you give me some resources for learning more about simplifying expressions with exponents?

A: Yes, here are some resources for learning more about simplifying expressions with exponents:

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Conclusion

In this article, we have discussed the power rule of exponents, how to simplify expressions with a variable base and an exponent, and how to get rid of negative exponents. We have also provided some practice problems and tips for simplifying expressions with exponents.