Which Expression Is Equivalent To $\left(\frac{m^5 N}{p Q^2}\right)^4$?A. $\frac{m^9 N^5}{p^5 Q^6}$ B. $\frac{m^{20} N^4}{p Q^2}$ C. $\frac{m^{20} N^4}{p^4 Q^8}$ D. $\frac{m^9 N^4}{p^4 Q^6}$

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Understanding Exponents and Equivalent Expressions

In mathematics, exponents are a fundamental concept used to represent repeated multiplication of a number. When dealing with exponents, it's essential to understand the rules of exponentiation, including the power of a power rule, the product of powers rule, and the quotient of powers rule. In this article, we will explore the concept of equivalent expressions and how to simplify complex expressions using exponent rules.

The Power of a Power Rule

The power of a power rule states that when raising a power to another power, we multiply the exponents. Mathematically, this can be represented as:

(am)n=am⋅n(a^m)^n = a^{m \cdot n}

The Product of Powers Rule

The product of powers rule states that when multiplying powers with the same base, we add the exponents. Mathematically, this can be represented as:

am⋅an=am+na^m \cdot a^n = a^{m + n}

The Quotient of Powers Rule

The quotient of powers rule states that when dividing powers with the same base, we subtract the exponents. Mathematically, this can be represented as:

aman=am−n\frac{a^m}{a^n} = a^{m - n}

Applying Exponent Rules to the Given Expression

Now that we have a solid understanding of the exponent rules, let's apply them to the given expression:

(m5npq2)4\left(\frac{m^5 n}{p q^2}\right)^4

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

(m5)4⋅(n)4(p)4⋅(q2)4\frac{(m^5)^4 \cdot (n)^4}{(p)^4 \cdot (q^2)^4}

Applying the product of powers rule, we can simplify the expression further:

m5⋅4⋅n4p4⋅q2⋅4\frac{m^{5 \cdot 4} \cdot n^4}{p^4 \cdot q^{2 \cdot 4}}

Simplifying the exponents, we get:

m20⋅n4p4⋅q8\frac{m^{20} \cdot n^4}{p^4 \cdot q^8}

Comparing the Simplified Expression to the Answer Choices

Now that we have simplified the expression, let's compare it to the answer choices:

A. m9n5p5q6\frac{m^9 n^5}{p^5 q^6} B. m20n4pq2\frac{m^{20} n^4}{p q^2} C. m20n4p4q8\frac{m^{20} n^4}{p^4 q^8} D. m9n4p4q6\frac{m^9 n^4}{p^4 q^6}

Based on our simplified expression, we can see that the correct answer is:

C. m20n4p4q8\frac{m^{20} n^4}{p^4 q^8}

Conclusion

In conclusion, we have successfully simplified the given expression using exponent rules and compared it to the answer choices. By applying the power of a power rule, the product of powers rule, and the quotient of powers rule, we were able to simplify the expression and determine the correct answer. This exercise demonstrates the importance of understanding exponent rules and how to apply them to simplify complex expressions.

Key Takeaways

  • The power of a power rule states that when raising a power to another power, we multiply the exponents.
  • The product of powers rule states that when multiplying powers with the same base, we add the exponents.
  • The quotient of powers rule states that when dividing powers with the same base, we subtract the exponents.
  • By applying exponent rules, we can simplify complex expressions and determine equivalent expressions.

Final Answer

Frequently Asked Questions

In the previous article, we explored the concept of equivalent expressions and how to simplify complex expressions using exponent rules. In this article, we will answer some frequently asked questions related to exponents and equivalent expressions.

Q: What is the power of a power rule?

A: The power of a power rule states that when raising a power to another power, we multiply the exponents. Mathematically, this can be represented as:

(am)n=am⋅n(a^m)^n = a^{m \cdot n}

Q: What is the product of powers rule?

A: The product of powers rule states that when multiplying powers with the same base, we add the exponents. Mathematically, this can be represented as:

am⋅an=am+na^m \cdot a^n = a^{m + n}

Q: What is the quotient of powers rule?

A: The quotient of powers rule states that when dividing powers with the same base, we subtract the exponents. Mathematically, this can be represented as:

aman=am−n\frac{a^m}{a^n} = a^{m - n}

Q: How do I apply exponent rules to simplify complex expressions?

A: To apply exponent rules, follow these steps:

  1. Identify the base and exponent in the expression.
  2. Apply the power of a power rule, product of powers rule, or quotient of powers rule as necessary.
  3. Simplify the expression by combining like terms.

Q: What is an equivalent expression?

A: An equivalent expression is an expression that has the same value as the original expression, but may be written in a different form. Equivalent expressions can be obtained by applying exponent rules, multiplying or dividing by the same non-zero number, or adding or subtracting the same value.

Q: How do I determine if two expressions are equivalent?

A: To determine if two expressions are equivalent, follow these steps:

  1. Simplify both expressions using exponent rules and other algebraic properties.
  2. Compare the simplified expressions to determine if they are equal.

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

A: Some common mistakes to avoid when working with exponents include:

  • Forgetting to apply the power of a power rule when raising a power to another power.
  • Forgetting to apply the product of powers rule when multiplying powers with the same base.
  • Forgetting to apply the quotient of powers rule when dividing powers with the same base.
  • Not simplifying expressions by combining like terms.

Q: How do I practice working with exponents and equivalent expressions?

A: To practice working with exponents and equivalent expressions, try the following:

  • Work through practice problems and exercises in your textbook or online resources.
  • Use online tools or calculators to check your work and identify areas for improvement.
  • Practice simplifying complex expressions using exponent rules and other algebraic properties.
  • Work with a study group or tutor to get help and feedback on your work.

Conclusion

In conclusion, we have answered some frequently asked questions related to exponents and equivalent expressions. By understanding exponent rules and how to apply them, you can simplify complex expressions and determine equivalent expressions. Remember to practice working with exponents and equivalent expressions to build your skills and confidence.

Key Takeaways

  • The power of a power rule states that when raising a power to another power, we multiply the exponents.
  • The product of powers rule states that when multiplying powers with the same base, we add the exponents.
  • The quotient of powers rule states that when dividing powers with the same base, we subtract the exponents.
  • Equivalent expressions can be obtained by applying exponent rules, multiplying or dividing by the same non-zero number, or adding or subtracting the same value.

Final Answer

The final answer is that by understanding exponent rules and how to apply them, you can simplify complex expressions and determine equivalent expressions.