Simplify The Expression:${ -3mn(m 2n 3 + 2mn) }$

Mar 8, 2025 by ADMIN 50 views

Simplify the Expression: $-3mn(m^2n^3 + 2mn)$

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it is essential to understand the rules and techniques involved in simplifying expressions. In this article, we will focus on simplifying the given expression: $-3mn(m^2n^3 + 2mn)$ . We will use various techniques such as the distributive property, combining like terms, and factoring to simplify the expression.

Understanding the Expression

The given expression is $-3mn(m^2n^3 + 2mn)$ . This expression consists of three terms: $-3mn$ , $m^2n^3$ , and $2mn$ . The first term is a constant term, while the second and third terms are variables. The expression is enclosed in parentheses, which indicates that it is a single term.

Distributive Property

The distributive property is a fundamental concept in algebra that states that for any real numbers $a$ , $b$ , and $c$ , the following equation holds:

$a(b + c) = ab + ac$

We can use the distributive property to simplify the given expression. To do this, we need to multiply the constant term $-3mn$ with each term inside the parentheses.

Applying the Distributive Property

Using the distributive property, we can rewrite the expression as:

$-3mn(m^2n^3 + 2mn) = -3mn \cdot m^2n^3 + (-3mn) \cdot 2mn$

Simplifying the Expression

Now that we have applied the distributive property, we can simplify the expression further. We can start by simplifying the first term:

$-3mn \cdot m^2n^3 = -3m^3n^4$

Next, we can simplify the second term:

$(-3mn) \cdot 2mn = -6m^2n^3$

Combining Like Terms

Now that we have simplified each term, we can combine like terms. The expression now becomes:

$-3m^3n^4 - 6m^2n^3$

Factoring

We can factor out the greatest common factor (GCF) from the expression. The GCF of $-3m^3n^4$ and $-6m^2n^3$ is $-3mn^3$ . We can factor out $-3mn^3$ from each term:

$-3mn^3(m^2 - 2n)$

Conclusion

In this article, we simplified the expression $-3mn(m^2n^3 + 2mn)$ using various techniques such as the distributive property, combining like terms, and factoring. We started by applying the distributive property to multiply the constant term with each term inside the parentheses. We then simplified each term and combined like terms. Finally, we factored out the greatest common factor from the expression. The simplified expression is $-3mn^3(m^2 - 2n)$ .

Final Answer

The final answer is $\boxed{-3mn^3(m^2 - 2n)}$ .

Step-by-Step Solution

Here is the step-by-step solution to simplify the expression:

Apply the distributive property to multiply the constant term with each term inside the parentheses.
Simplify each term.
Combine like terms.
Factor out the greatest common factor from the expression.

Common Mistakes

When simplifying expressions, it is essential to avoid common mistakes such as:

Not applying the distributive property correctly.
Not simplifying each term correctly.
Not combining like terms correctly.
Not factoring out the greatest common factor correctly.

Tips and Tricks

Here are some tips and tricks to help you simplify expressions:

Always apply the distributive property correctly.
Simplify each term step by step.
Combine like terms carefully.
Factor out the greatest common factor correctly.

Real-World Applications

Simplifying expressions has many real-world applications, such as:

Solving systems of equations.
Finding the area and perimeter of shapes.
Calculating the volume of solids.
Modeling real-world phenomena.

Conclusion

In conclusion, simplifying expressions is a crucial skill in mathematics, and it is essential to understand the rules and techniques involved in simplifying expressions. By applying the distributive property, combining like terms, and factoring, we can simplify expressions and solve problems in various fields.

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Introduction

In our previous article, we simplified the expression $-3mn(m^2n^3 + 2mn)$ using various techniques such as the distributive property, combining like terms, and factoring. In this article, we will answer some frequently asked questions (FAQs) related to simplifying expressions.

Q&A

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that states that for any real numbers $a$ , $b$ , and $c$ , the following equation holds:

$a(b + c) = ab + ac$

This property allows us to multiply a constant term with each term inside the parentheses.

Q: How do I apply the distributive property?

A: To apply the distributive property, you need to multiply the constant term with each term inside the parentheses. For example, if you have the expression $2(x + 3)$ , you can apply the distributive property by multiplying 2 with each term inside the parentheses:

$2(x + 3) = 2x + 6$

Q: What is the difference between combining like terms and factoring?

A: Combining like terms involves adding or subtracting terms that have the same variable and exponent. Factoring involves expressing an expression as a product of simpler expressions.

For example, consider the expression $x^2 + 4x + 4$ . We can combine like terms by adding the terms:

$x^2 + 4x + 4 = x^2 + 4x + 4$

However, we can also factor the expression as:

$x^2 + 4x + 4 = (x + 2)^2$

Q: How do I factor an expression?

A: To factor an expression, you need to find the greatest common factor (GCF) of the terms and express the expression as a product of simpler expressions. For example, consider the expression $6x^2 + 12x$ . We can factor out the GCF, which is 6x:

$6x^2 + 12x = 6x(x + 2)$

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

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

Not applying the distributive property correctly.
Not simplifying each term correctly.
Not combining like terms correctly.
Not factoring out the greatest common factor correctly.

Q: How do I know when to use the distributive property and when to factor?

A: You should use the distributive property when you have an expression with a constant term and a variable term inside the parentheses. You should factor when you have an expression that can be expressed as a product of simpler expressions.

Q: Can you give an example of a real-world application of simplifying expressions?

A: Yes, simplifying expressions has many real-world applications, such as:

Solving systems of equations.
Finding the area and perimeter of shapes.
Calculating the volume of solids.
Modeling real-world phenomena.

For example, consider a company that wants to calculate the cost of producing a certain number of units. The cost can be expressed as an expression, and simplifying the expression can help the company calculate the cost more easily.

Conclusion

Final Answer

The final answer is $\boxed{-3mn^3(m^2 - 2n)}$ .

Step-by-Step Solution

Here is the step-by-step solution to simplify the expression:

Apply the distributive property to multiply the constant term with each term inside the parentheses.
Simplify each term.
Combine like terms.
Factor out the greatest common factor from the expression.

Common Mistakes

When simplifying expressions, it is essential to avoid common mistakes such as:

Not applying the distributive property correctly.
Not simplifying each term correctly.
Not combining like terms correctly.
Not factoring out the greatest common factor correctly.

Tips and Tricks

Here are some tips and tricks to help you simplify expressions:

Always apply the distributive property correctly.
Simplify each term step by step.
Combine like terms carefully.
Factor out the greatest common factor correctly.

Real-World Applications

Simplifying expressions has many real-world applications, such as:

Solving systems of equations.
Finding the area and perimeter of shapes.
Calculating the volume of solids.
Modeling real-world phenomena.