4. To Simplify The Expression $\left(\frac{3^4 M^2 Y^3}{5^2 M^{-3}}\right)^2$, The First Step Would Be To:A. Simplify $3^4$ And $5^2$B. Apply The Quotient Rule To $m^2$ And $m^{-3}$C. Apply The Power Rule

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Understanding the Problem

When simplifying complex exponential expressions, it's essential to apply the correct rules to ensure accuracy. In this article, we'll focus on simplifying the expression (34m2y352m−3)2\left(\frac{3^4 m^2 y^3}{5^2 m^{-3}}\right)^2 using the power rule.

The Power Rule: A Key Concept

The power rule is a fundamental concept in algebra that states: (am)n=amn(a^m)^n = a^{mn}. This rule allows us to simplify expressions by multiplying the exponents. In the given expression, we need to apply the power rule to simplify the exponent of the entire fraction.

Applying the Power Rule

To simplify the expression (34m2y352m−3)2\left(\frac{3^4 m^2 y^3}{5^2 m^{-3}}\right)^2, we'll start by applying the power rule. According to the power rule, we can rewrite the expression as:

(34m2y352m−3)2=(34)2(m2)2(y3)2(52)2(m−3)2\left(\frac{3^4 m^2 y^3}{5^2 m^{-3}}\right)^2 = \frac{(3^4)^2 (m^2)^2 (y^3)^2}{(5^2)^2 (m^{-3})^2}

Simplifying the Exponents

Now that we've applied the power rule, we can simplify the exponents. Using the power rule, we can rewrite each exponent as:

(34)2=34⋅2=38(3^4)^2 = 3^{4 \cdot 2} = 3^8

(m2)2=m2⋅2=m4(m^2)^2 = m^{2 \cdot 2} = m^4

(y3)2=y3⋅2=y6(y^3)^2 = y^{3 \cdot 2} = y^6

(52)2=52⋅2=54(5^2)^2 = 5^{2 \cdot 2} = 5^4

(m−3)2=m−3⋅2=m−6(m^{-3})^2 = m^{-3 \cdot 2} = m^{-6}

Simplifying the Expression

Now that we've simplified the exponents, we can rewrite the expression as:

38m4y654m−6\frac{3^8 m^4 y^6}{5^4 m^{-6}}

Applying the Quotient Rule

To simplify the expression further, we can apply the quotient rule, which states: aman=am−n\frac{a^m}{a^n} = a^{m-n}. In this case, we can rewrite the expression as:

38m4y654m−6=38m4+6y654\frac{3^8 m^4 y^6}{5^4 m^{-6}} = \frac{3^8 m^{4+6} y^6}{5^4}

Simplifying the Expression

Now that we've applied the quotient rule, we can simplify the expression further:

38m4+6y654=38m10y654\frac{3^8 m^{4+6} y^6}{5^4} = \frac{3^8 m^{10} y^6}{5^4}

Conclusion

In this article, we've demonstrated how to simplify the expression (34m2y352m−3)2\left(\frac{3^4 m^2 y^3}{5^2 m^{-3}}\right)^2 using the power rule and the quotient rule. By applying these rules, we were able to simplify the expression and arrive at the final answer.

Key Takeaways

  • The power rule states: (am)n=amn(a^m)^n = a^{mn}.
  • The quotient rule states: aman=am−n\frac{a^m}{a^n} = a^{m-n}.
  • To simplify complex exponential expressions, it's essential to apply the correct rules.
  • The power rule and the quotient rule are essential concepts in algebra that can be used to simplify expressions.

Final Answer

The final answer is: 38m10y654\boxed{\frac{3^8 m^{10} y^6}{5^4}}

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Understanding the Problem

In our previous article, we demonstrated how to simplify the expression (34m2y352m−3)2\left(\frac{3^4 m^2 y^3}{5^2 m^{-3}}\right)^2 using the power rule and the quotient rule. In this article, we'll provide a Q&A guide to help you better understand the concepts and apply them to similar problems.

Q: What is the power rule?

A: The power rule is a fundamental concept in algebra that states: (am)n=amn(a^m)^n = a^{mn}. This rule allows us to simplify expressions by multiplying the exponents.

Q: How do I apply the power rule?

A: To apply the power rule, simply multiply the exponents of the base. For example, if we have (34)2(3^4)^2, we can rewrite it as 34⋅2=383^{4 \cdot 2} = 3^8.

Q: What is the quotient rule?

A: The quotient rule is another fundamental concept in algebra that states: aman=am−n\frac{a^m}{a^n} = a^{m-n}. This rule allows us to simplify expressions by subtracting the exponents.

Q: How do I apply the quotient rule?

A: To apply the quotient rule, simply subtract the exponents of the base. For example, if we have 3432\frac{3^4}{3^2}, we can rewrite it as 34−2=323^{4-2} = 3^2.

Q: Can I apply the power rule and the quotient rule together?

A: Yes, you can apply the power rule and the quotient rule together to simplify complex exponential expressions. For example, if we have (34m2y352m−3)2\left(\frac{3^4 m^2 y^3}{5^2 m^{-3}}\right)^2, we can apply the power rule first and then the quotient rule.

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

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

  • Not applying the power rule or the quotient rule correctly
  • Not simplifying the exponents correctly
  • Not canceling out common factors
  • Not checking the final answer for accuracy

Q: How can I practice simplifying exponential expressions?

A: You can practice simplifying exponential expressions by working through examples and exercises. You can also try simplifying expressions on your own and then checking your answers with a calculator or a teacher.

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

A: Simplifying exponential expressions has many real-world applications, including:

  • Calculating interest rates and investments
  • Modeling population growth and decay
  • Analyzing data and statistics
  • Solving problems in physics and engineering

Conclusion

In this article, we've provided a Q&A guide to help you better understand the concepts of simplifying exponential expressions. By applying the power rule and the quotient rule, you can simplify complex expressions and arrive at the final answer.

Key Takeaways

  • The power rule states: (am)n=amn(a^m)^n = a^{mn}.
  • The quotient rule states: aman=am−n\frac{a^m}{a^n} = a^{m-n}.
  • To simplify complex exponential expressions, it's essential to apply the correct rules.
  • Practice is key to mastering the concepts of simplifying exponential expressions.

Final Answer

The final answer is: Simplifying exponential expressions is a crucial skill in algebra and has many real-world applications. By applying the power rule and the quotient rule, you can simplify complex expressions and arrive at the final answer.