Simplify: 6 M 2 N 2 − 3 M N + 14 M N + 10 M 2 N 2 + 9 6m^2n^2 - 3mn + 14mn + 10m^2n^2 + 9 6 M 2 N 2 − 3 Mn + 14 Mn + 10 M 2 N 2 + 9 A. 27 M 2 N 2 + 9 27m^2n^2 + 9 27 M 2 N 2 + 9 B. 16 M 2 N 2 + 17 M N + 9 16m^2n^2 + 17mn + 9 16 M 2 N 2 + 17 Mn + 9 C. 16 M 2 N 2 + 11 M N + 9 16m^2n^2 + 11mn + 9 16 M 2 N 2 + 11 Mn + 9 D. 16 M 4 N 4 + 11 M 2 N 2 + 9 16m^4n^4 + 11m^2n^2 + 9 16 M 4 N 4 + 11 M 2 N 2 + 9

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for students and professionals alike. In this article, we will focus on simplifying a given algebraic expression, which involves combining like terms and applying the rules of arithmetic operations.

The Given Expression

The given expression is:

6m2n2−3mn+14mn+10m2n2+96m^2n^2 - 3mn + 14mn + 10m^2n^2 + 9

Our goal is to simplify this expression by combining like terms and eliminating any unnecessary components.

Step 1: Identify Like Terms

Like terms are terms that have the same variable(s) raised to the same power. In the given expression, we can identify the following like terms:

  • 6m2n26m^2n^2 and 10m2n210m^2n^2 (both have m2n2m^2n^2 as the variable)
  • −3mn-3mn and 14mn14mn (both have mnmn as the variable)

Step 2: Combine Like Terms

Now that we have identified the like terms, we can combine them by adding or subtracting their coefficients.

  • 6m2n2+10m2n2=16m2n26m^2n^2 + 10m^2n^2 = 16m^2n^2
  • −3mn+14mn=11mn-3mn + 14mn = 11mn

Step 3: Rewrite the Expression

Using the results from the previous step, we can rewrite the expression as:

16m2n2+11mn+916m^2n^2 + 11mn + 9

Step 4: Check for Any Other Simplifications

At this point, we have simplified the expression as much as possible. However, we can still check if there are any other simplifications that can be made.

Conclusion

In conclusion, the simplified expression is:

16m2n2+11mn+916m^2n^2 + 11mn + 9

This expression cannot be simplified further, and it is the final answer.

Comparison with Answer Choices

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

  • A. 27m2n2+927m^2n^2 + 9
  • B. 16m2n2+17mn+916m^2n^2 + 17mn + 9
  • C. 16m2n2+11mn+916m^2n^2 + 11mn + 9
  • D. 16m4n4+11m2n2+916m^4n^4 + 11m^2n^2 + 9

The correct answer is C. 16m2n2+11mn+916m^2n^2 + 11mn + 9, which matches our simplified expression.

Final Answer

The final answer is C. 16m2n2+11mn+916m^2n^2 + 11mn + 9.

Tips and Tricks

  • When simplifying algebraic expressions, it's essential to identify like terms and combine them.
  • Use the distributive property to expand expressions and simplify them.
  • Check for any other simplifications that can be made after combining like terms.

Common Mistakes

  • Failing to identify like terms and combine them.
  • Not using the distributive property to expand expressions.
  • Not checking for any other simplifications that can be made.

Real-World Applications

Simplifying algebraic expressions has numerous real-world applications, including:

  • Physics: Simplifying expressions is essential in physics to solve problems related to motion, energy, and forces.
  • Engineering: Simplifying expressions is crucial in engineering to design and optimize systems.
  • Computer Science: Simplifying expressions is necessary in computer science to develop efficient algorithms and data structures.

Conclusion

In conclusion, simplifying algebraic expressions is a fundamental skill that has numerous real-world applications. By identifying like terms, combining them, and applying the rules of arithmetic operations, we can simplify expressions and arrive at the final answer.

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Introduction

In our previous article, we discussed how to simplify algebraic expressions by combining like terms and applying the rules of arithmetic operations. In this article, we will provide a Q&A guide to help you understand the concepts and techniques involved in simplifying algebraic expressions.

Q1: What are like terms in algebraic expressions?

A1: Like terms are terms that have the same variable(s) raised to the same power. For example, 2x22x^2 and 5x25x^2 are like terms because they both have x2x^2 as the variable.

Q2: How do I identify like terms in an algebraic expression?

A2: To identify like terms, look for terms that have the same variable(s) raised to the same power. You can also use the distributive property to expand expressions and simplify them.

Q3: What is the distributive property in algebra?

A3: The distributive property is a rule that allows you to expand expressions by multiplying each term inside the parentheses by the term outside the parentheses. For example, a(b+c)=ab+aca(b + c) = ab + ac.

Q4: How do I combine like terms in an algebraic expression?

A4: To combine like terms, add or subtract their coefficients. For example, 2x2+5x2=7x22x^2 + 5x^2 = 7x^2.

Q5: What are some common mistakes to avoid when simplifying algebraic expressions?

A5: Some common mistakes to avoid include:

  • Failing to identify like terms and combine them.
  • Not using the distributive property to expand expressions.
  • Not checking for any other simplifications that can be made.

Q6: How do I check for any other simplifications that can be made after combining like terms?

A6: To check for any other simplifications, look for any common factors that can be factored out of the expression. You can also use the distributive property to expand expressions and simplify them.

Q7: What are some real-world applications of simplifying algebraic expressions?

A7: Simplifying algebraic expressions has numerous real-world applications, including:

  • Physics: Simplifying expressions is essential in physics to solve problems related to motion, energy, and forces.
  • Engineering: Simplifying expressions is crucial in engineering to design and optimize systems.
  • Computer Science: Simplifying expressions is necessary in computer science to develop efficient algorithms and data structures.

Q8: How do I simplify expressions with negative coefficients?

A8: To simplify expressions with negative coefficients, combine like terms as usual, but be careful with the signs. For example, −2x2+5x2=3x2-2x^2 + 5x^2 = 3x^2.

Q9: What is the difference between simplifying an expression and solving an equation?

A9: Simplifying an expression involves combining like terms and applying the rules of arithmetic operations to arrive at a simpler expression. Solving an equation involves finding the value of the variable that makes the equation true.

Q10: How do I know when an expression is simplified?

A10: An expression is simplified when there are no like terms left to combine, and the expression cannot be further simplified by factoring or applying the distributive property.

Conclusion

In conclusion, simplifying algebraic expressions is a fundamental skill that has numerous real-world applications. By identifying like terms, combining them, and applying the rules of arithmetic operations, we can simplify expressions and arrive at the final answer. We hope this Q&A guide has helped you understand the concepts and techniques involved in simplifying algebraic expressions.