Simplify The Following Monomials:${ \begin{array}{l} 11c 2d 2 - 20c 2d 2 \ -9c 2d 2 \quad -9c 4d 4 \end{array} }$ { -5cd \quad -5c^4d^4 \}

Mar 13, 2025 by ADMIN 141 views

Simplifying Monomials: A Step-by-Step Guide

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Introduction

Monomials are a fundamental concept in algebra, and simplifying them is an essential skill for any math enthusiast. In this article, we will explore the process of simplifying monomials, focusing on the given expression: ${ \begin{array}{l} 11c^2d2 - 20c^2d2 \ -9c^2d2 \quad -9c^4d4 \end{array} }$ ${ -5cd \quad -5c^4d^4 \}$ $

Understanding Monomials

A monomial is an algebraic expression consisting of a single term, which can be a number, a variable, or a product of variables and numbers. Monomials can be added, subtracted, multiplied, and divided, just like regular numbers. However, when simplifying monomials, we need to follow specific rules to ensure that the resulting expression is in its simplest form.

Simplifying the Given Expression

To simplify the given expression, we need to combine like terms. Like terms are monomials that have the same variable(s) raised to the same power(s). In this case, we have several like terms that we can combine.

Combining Like Terms

Let's start by combining the like terms in the first row of the given expression:

{ \begin{array}{l} 11c^2d^2 - 20c^2d^2 \\ -9c^2d^2 \quad -9c^4d^4 \end{array} \}

We can see that the first two terms, $11c^2d^2$ and $-20c^2d^2$ , are like terms because they both have the same variable(s) raised to the same power(s). To combine these terms, we add their coefficients:

$11c^2d^2 - 20c^2d^2 = -9c^2d^2$

Now, let's combine the like terms in the second row of the given expression:

$-9c^2d^2 \quad -9c^4d^4$

We can see that the first term, $-9c^2d^2$ , is not like the second term, $-9c^4d^4$ , because they have different variable(s) raised to different power(s). Therefore, we cannot combine these terms.

Combining the Remaining Terms

Now, let's combine the remaining terms in the given expression:

$-5cd \quad -5c^4d^4$

We can see that the first term, $-5cd$ , is not like the second term, $-5c^4d^4$ , because they have different variable(s) raised to different power(s). Therefore, we cannot combine these terms.

Final Simplified Expression

After combining like terms, we are left with the following simplified expression:

$-9c^2d^2 - 9c^4d^4 - 5cd$

This is the simplest form of the given expression.

Conclusion

Simplifying monomials is an essential skill for any math enthusiast. By following the rules of combining like terms, we can simplify complex expressions and arrive at their simplest form. In this article, we explored the process of simplifying monomials, focusing on the given expression. We combined like terms, identified non-like terms, and arrived at the final simplified expression.

Frequently Asked Questions

Q: What are like terms?

A: Like terms are monomials that have the same variable(s) raised to the same power(s).

Q: How do I combine like terms?

A: To combine like terms, add their coefficients.

Q: What if I have non-like terms?

A: If you have non-like terms, you cannot combine them.

Q: How do I simplify a monomial expression?

A: To simplify a monomial expression, combine like terms and identify non-like terms.

Additional Resources

For more information on simplifying monomials, check out the following resources:

Khan Academy: Simplifying Monomials
Mathway: Simplifying Monomials
Wolfram Alpha: Simplifying Monomials

Final Thoughts

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Introduction

In our previous article, we explored the process of simplifying monomials, focusing on the given expression: ${ \begin{array}{l} 11c^2d2 - 20c^2d2 \ -9c^2d2 \quad -9c^4d4 \end{array} }$ ${ -5cd \quad -5c^4d^4 \}$ $

In this article, we will answer some of the most frequently asked questions about simplifying monomials.

Q&A

Q: What are like terms?

A: Like terms are monomials that have the same variable(s) raised to the same power(s). For example, $2x^2$ and $3x^2$ are like terms because they both have the variable $x$ raised to the power of 2.

Q: How do I combine like terms?

A: To combine like terms, add their coefficients. For example, if we have the expression $2x^2 + 3x^2$ , we can combine the like terms by adding their coefficients: $2x^2 + 3x^2 = 5x^2$ .

Q: What if I have non-like terms?

A: If you have non-like terms, you cannot combine them. For example, if we have the expression $2x^2 + 3y^2$ , we cannot combine the like terms because they have different variables.

Q: How do I simplify a monomial expression?

A: To simplify a monomial expression, combine like terms and identify non-like terms. For example, if we have the expression $2x^2 + 3x^2 - 4x^2$ , we can simplify it by combining the like terms: $2x^2 + 3x^2 - 4x^2 = x^2$ .

Q: Can I simplify a monomial expression with variables raised to different powers?

A: Yes, you can simplify a monomial expression with variables raised to different powers. For example, if we have the expression $2x^2 + 3x^3$ , we can simplify it by combining the like terms: $2x^2 + 3x^3 = 3x^3 + 2x^2$ .

Q: How do I simplify a monomial expression with negative coefficients?

A: To simplify a monomial expression with negative coefficients, combine the like terms and change the sign of the result. For example, if we have the expression $-2x^2 + 3x^2$ , we can simplify it by combining the like terms: $-2x^2 + 3x^2 = x^2$ .

Q: Can I simplify a monomial expression with variables raised to different powers and negative coefficients?

A: Yes, you can simplify a monomial expression with variables raised to different powers and negative coefficients. For example, if we have the expression $-2x^2 + 3x^3$ , we can simplify it by combining the like terms: $-2x^2 + 3x^3 = 3x^3 - 2x^2$ .