Choose The Correctly Simplified Expression. 42 M − 3 N 0 6 M 8 N − 2 \frac{42 M^{-3} N^0}{6 M^8 N^{-2}} 6 M 8 N − 2 42 M − 3 N 0 ​ A. 7 6 M 11 N 2 \frac{7}{6 M^{11} N^2} 6 M 11 N 2 7 ​ B. 7 M 11 2 N 3 \frac{7 M^{11}}{2 N^3} 2 N 3 7 M 11 ​ C. 7 M 11 N 2 \frac{7 M^{11}}{n^2} N 2 7 M 11 ​ D. 7 N 2 M 11 \frac{7 N^2}{m^{11}} M 11 7 N 2 ​

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Understanding Exponents and Their Rules

Exponents are a fundamental concept in algebra, and understanding how to simplify expressions with exponents is crucial for solving mathematical problems. In this article, we will explore the rules of exponents and how to apply them to simplify algebraic expressions.

The Rules of Exponents

Before we dive into simplifying the given expression, let's review the rules of exponents:

  • Product Rule: When multiplying two or more variables with the same base, add the exponents. For example, am⋅an=am+na^m \cdot a^n = a^{m+n}.
  • Quotient Rule: When dividing two or more variables with the same base, subtract the exponents. For example, aman=am−n\frac{a^m}{a^n} = a^{m-n}.
  • Power Rule: When raising a variable with an exponent to another power, multiply the exponents. For example, (am)n=am⋅n(a^m)^n = a^{m \cdot n}.
  • Zero Exponent Rule: Any non-zero variable raised to the power of zero is equal to 1. For example, a0=1a^0 = 1.

Simplifying the Given Expression

Now that we have reviewed the rules of exponents, let's simplify the given expression:

42m−3n06m8n−2\frac{42 m^{-3} n^0}{6 m^8 n^{-2}}

To simplify this expression, we need to apply the rules of exponents. First, let's simplify the numerator:

42m−3n0=42⋅m−3⋅n042 m^{-3} n^0 = 42 \cdot m^{-3} \cdot n^0

Since n0=1n^0 = 1, we can simplify the expression further:

42⋅m−3⋅n0=42⋅m−342 \cdot m^{-3} \cdot n^0 = 42 \cdot m^{-3}

Now, let's simplify the denominator:

6m8n−2=6⋅m8⋅n−26 m^8 n^{-2} = 6 \cdot m^8 \cdot n^{-2}

To simplify the expression further, we need to apply the quotient rule:

42⋅m−36⋅m8⋅n−2=426⋅m−3m8⋅n21\frac{42 \cdot m^{-3}}{6 \cdot m^8 \cdot n^{-2}} = \frac{42}{6} \cdot \frac{m^{-3}}{m^8} \cdot \frac{n^2}{1}

Now, let's simplify the fractions:

426=7\frac{42}{6} = 7

m−3m8=m−3−8=m−11\frac{m^{-3}}{m^8} = m^{-3-8} = m^{-11}

n21=n2\frac{n^2}{1} = n^2

So, the simplified expression is:

7⋅m−11⋅n27 \cdot m^{-11} \cdot n^2

Choosing the Correct Answer

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

A. 76m11n2\frac{7}{6 m^{11} n^2} B. 7m112n3\frac{7 m^{11}}{2 n^3} C. 7m11n2\frac{7 m^{11}}{n^2} D. 7n2m11\frac{7 n^2}{m^{11}}

The correct answer is C. 7m11n2\frac{7 m^{11}}{n^2}.

Conclusion

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent indicates that the base is raised to a power, while a negative exponent indicates that the base is raised to a power and then taken as a reciprocal. For example, ama^m is different from a−ma^{-m}.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, you need to apply the quotient rule. For example, aman=am−n\frac{a^m}{a^n} = a^{m-n}. If the exponent is negative, you can rewrite it as a positive exponent by taking the reciprocal of the base. For example, a−m=1ama^{-m} = \frac{1}{a^m}.

Q: What is the rule for multiplying variables with exponents?

A: When multiplying variables with exponents, you need to add the exponents. For example, am⋅an=am+na^m \cdot a^n = a^{m+n}.

Q: What is the rule for dividing variables with exponents?

A: When dividing variables with exponents, you need to subtract the exponents. For example, aman=am−n\frac{a^m}{a^n} = a^{m-n}.

Q: How do I simplify an expression with a zero exponent?

A: Any non-zero variable raised to the power of zero is equal to 1. For example, a0=1a^0 = 1.

Q: What is the rule for raising a variable with an exponent to another power?

A: When raising a variable with an exponent to another power, you need to multiply the exponents. For example, (am)n=am⋅n(a^m)^n = a^{m \cdot n}.

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

A: To simplify an expression with multiple variables and exponents, you need to apply the rules of exponents in the correct order. First, simplify the expression inside the parentheses, then apply the product rule, quotient rule, and power rule as needed.

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

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

  • Not applying the rules of exponents in the correct order
  • Not simplifying the expression inside the parentheses
  • Not using the correct signs for positive and negative exponents
  • Not applying the quotient rule when dividing variables with exponents

Q: How can I practice simplifying exponents?

A: You can practice simplifying exponents by working through examples and exercises in a textbook or online resource. You can also try simplifying expressions with exponents on your own, using a calculator or computer program to check your work.

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

A: Simplifying exponents has many real-world applications, including:

  • Physics and engineering: Exponents are used to describe the behavior of physical systems, such as the motion of objects and the flow of fluids.
  • Computer science: Exponents are used in algorithms and data structures, such as binary search trees and hash tables.
  • Economics: Exponents are used to model economic systems, such as the behavior of markets and the growth of economies.

Conclusion

Simplifying exponents is an essential skill in mathematics and has many real-world applications. By understanding the rules of exponents and practicing simplifying expressions, you can become proficient in this skill and apply it to a wide range of problems.