Simplify The Expression:$\[ 11(4cd)\left(-cd^5\right) \\]

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the rules and techniques involved. In this article, we will focus on simplifying the given expression: 11(4cd)(−cd5)11(4cd)\left(-cd^5\right). We will break down the expression step by step, using the rules of algebra to simplify it.

Understanding the Expression

The given expression is a product of three terms: 1111, 4cd4cd, and −cd5-cd^5. To simplify the expression, we need to apply the rules of algebra, which include the distributive property, the commutative property, and the associative property.

Distributive Property

The distributive property states that for any numbers aa, bb, and cc, the following equation holds:

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

We can use this property to simplify the expression by distributing the 1111 to each term inside the parentheses.

Commutative Property

The commutative property states that for any numbers aa and bb, the following equation holds:

a+b=b+aa + b = b + a

We can use this property to rearrange the terms inside the parentheses.

Associative Property

The associative property states that for any numbers aa, bb, and cc, the following equation holds:

(a+b)+c=a+(b+c)(a + b) + c = a + (b + c)

We can use this property to group the terms inside the parentheses.

Simplifying the Expression

Now that we have a good understanding of the rules of algebra, we can start simplifying the expression.

Step 1: Distribute the 11

We can start by distributing the 1111 to each term inside the parentheses:

11(4cd)(−cd5)=11⋅4cd⋅−cd511(4cd)\left(-cd^5\right) = 11 \cdot 4cd \cdot -cd^5

Step 2: Simplify the Terms

We can simplify the terms inside the parentheses by multiplying the numbers and combining like terms:

11⋅4cd⋅−cd5=−44cd611 \cdot 4cd \cdot -cd^5 = -44cd^6

Step 3: Final Simplification

We have now simplified the expression to its final form:

−44cd6-44cd^6

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the rules and techniques involved. In this article, we have focused on simplifying the given expression: 11(4cd)(−cd5)11(4cd)\left(-cd^5\right). We have broken down the expression step by step, using the rules of algebra to simplify it. The final simplified expression is −44cd6-44cd^6.

Frequently Asked Questions

Q: What is the distributive property?

A: The distributive property is a rule in algebra that states that for any numbers aa, bb, and cc, the following equation holds: a(b+c)=ab+aca(b+c) = ab + ac.

Q: What is the commutative property?

A: The commutative property is a rule in algebra that states that for any numbers aa and bb, the following equation holds: a+b=b+aa + b = b + a.

Q: What is the associative property?

A: The associative property is a rule in algebra that states that for any numbers aa, bb, and cc, the following equation holds: (a+b)+c=a+(b+c)(a + b) + c = a + (b + c).

Further Reading

If you want to learn more about simplifying algebraic expressions, we recommend checking out the following resources:

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Q&A: Simplifying Algebraic Expressions

Q: What is the distributive property?

A: The distributive property is a rule in algebra that states that for any numbers aa, bb, and cc, the following equation holds: a(b+c)=ab+aca(b+c) = ab + ac. This property allows us to distribute a single term to multiple terms inside parentheses.

Q: What is the commutative property?

A: The commutative property is a rule in algebra that states that for any numbers aa and bb, the following equation holds: a+b=b+aa + b = b + a. This property allows us to rearrange the terms inside parentheses.

Q: What is the associative property?

A: The associative property is a rule in algebra that states that for any numbers aa, bb, and cc, the following equation holds: (a+b)+c=a+(b+c)(a + b) + c = a + (b + c). This property allows us to group the terms inside parentheses.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you need to apply the rules of algebra, including the distributive property, the commutative property, and the associative property. You can start by distributing a single term to multiple terms inside parentheses, then simplify the terms by combining like terms.

Q: What is the difference between a variable and a constant?

A: A variable is a letter or symbol that represents a value that can change, while a constant is a value that remains the same. In the expression 11(4cd)(−cd5)11(4cd)\left(-cd^5\right), the variables are cc and dd, while the constants are 1111 and 44.

Q: How do I combine like terms?

A: To combine like terms, you need to identify the terms that have the same variable and coefficient. You can then add or subtract the coefficients of the like terms to simplify the expression.

Q: What is the final simplified expression for 11(4cd)(−cd5)11(4cd)\left(-cd^5\right)?

A: The final simplified expression for 11(4cd)(−cd5)11(4cd)\left(-cd^5\right) is −44cd6-44cd^6.

Q: Can you provide more examples of simplifying algebraic expressions?

A: Yes, here are a few more examples:

  • Simplify the expression: 2(3x2)(4x3)2(3x^2)\left(4x^3\right)
  • Simplify the expression: 5(2y4)(−3y2)5(2y^4)\left(-3y^2\right)
  • Simplify the expression: 3(4z5)(−2z3)3(4z^5)\left(-2z^3\right)

Q: How do I apply the distributive property to simplify an algebraic expression?

A: To apply the distributive property, you need to distribute a single term to multiple terms inside parentheses. For example, in the expression 2(3x2)(4x3)2(3x^2)\left(4x^3\right), you can distribute the 22 to the terms inside the parentheses to get 2⋅3x2⋅4x32 \cdot 3x^2 \cdot 4x^3.

Q: How do I apply the commutative property to simplify an algebraic expression?

A: To apply the commutative property, you need to rearrange the terms inside parentheses. For example, in the expression 2(3x2)(4x3)2(3x^2)\left(4x^3\right), you can rearrange the terms to get 2⋅4x3⋅3x22 \cdot 4x^3 \cdot 3x^2.

Q: How do I apply the associative property to simplify an algebraic expression?

A: To apply the associative property, you need to group the terms inside parentheses. For example, in the expression 2(3x2)(4x3)2(3x^2)\left(4x^3\right), you can group the terms to get (2⋅3x2)⋅4x3(2 \cdot 3x^2) \cdot 4x^3.

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the rules and techniques involved. In this article, we have focused on simplifying the given expression: 11(4cd)(−cd5)11(4cd)\left(-cd^5\right). We have broken down the expression step by step, using the rules of algebra to simplify it. The final simplified expression is −44cd6-44cd^6. We have also provided additional examples and explanations to help you understand the concepts better.

Frequently Asked Questions

Q: What is the distributive property?

A: The distributive property is a rule in algebra that states that for any numbers aa, bb, and cc, the following equation holds: a(b+c)=ab+aca(b+c) = ab + ac.

Q: What is the commutative property?

A: The commutative property is a rule in algebra that states that for any numbers aa and bb, the following equation holds: a+b=b+aa + b = b + a.

Q: What is the associative property?

A: The associative property is a rule in algebra that states that for any numbers aa, bb, and cc, the following equation holds: (a+b)+c=a+(b+c)(a + b) + c = a + (b + c).

Further Reading

If you want to learn more about simplifying algebraic expressions, we recommend checking out the following resources: