8.3.2 - Multiplying three or more monomials

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Introduction to Multiplying Monomials

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Teacher
Teacher

Today we'll explore how to multiply three or more monomials. Can anyone remind me what a monomial is?

Student 1
Student 1

Isn't it an expression with just one term?

Teacher
Teacher

That's correct! A monomial has one term, like 2x or 3xy. Now, when we multiply them, what must we remember?

Student 2
Student 2

We need to multiply the coefficients and add the exponents for like bases!

Teacher
Teacher

Exactly! Let's use an example: 2x × 5y × 7z. What do we multiply first?

Student 3
Student 3

We multiply the first two: 2 × 5 = 10, and then add the variables.

Teacher
Teacher

Right! So, we have 10xy. Next, we multiply that by 7z. What's the final result?

Student 4
Student 4

It would be 70xyz!

Teacher
Teacher

Perfect! So our first takeaway is understanding how to organize and group our multiplication.

Multiplying Multiple Monomials

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Teacher
Teacher

Now, let's tackle multiplying more complex monomials. How about 4xy × 5x²y² × 6x³y³?

Student 1
Student 1

Do we start with the coefficients, like always?

Teacher
Teacher

That's right! So, what does 4 × 5 × 6 equal?

Student 2
Student 2

That’s 120.

Teacher
Teacher

Great! Now, how do we handle the variables?

Student 3
Student 3

We add the exponents for the same bases.

Teacher
Teacher

Correct! So what do we get for x and y?

Student 4
Student 4

It’s 120x^(1+2+3)y^(1+2+3) = 120x^6y^6!

Teacher
Teacher

Excellent! You've grasped it well! Key is recognizing the powers and coefficients.

Application in Geometry

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Teacher
Teacher

Now, how do you think we can apply multiplying monomials in real life?

Student 1
Student 1

In finding areas and volumes!

Teacher
Teacher

Absolutely! If I say the length is 2x, breadth is 3y, and height is 4z, how would we find the volume?

Student 2
Student 2

That would be 2x × 3y × 4z = 24xyz!

Teacher
Teacher

Exactly! So, remember, multiplying monomials helps us solve practical problems in geometry!

Introduction & Overview

Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.

Quick Overview

This section explains how to multiply three or more monomials together, illustrating the process with various examples.

Standard

In this section, students learn the systematic approach to multiplying three or more monomials, including the importance of grouping and combining like terms. The section provides detailed examples to solidify understanding, reinforcing the concepts of coefficients, variables, and powers.

Detailed

Multiplying Three or More Monomials

In this section, we extend our understanding of multiplication from two monomials to three or more. The multiplication of monomials involves multiplying their coefficients and adding the exponents of like bases. This section highlights the associative property of multiplication, allowing us to group terms in any order for ease of calculation.

For example, when multiplying the monomials 2x, 5y, and 7z, we first pair and multiply:

  1. Multiply the first two: (2x × 5y) = 10xy.
  2. Then multiply the result by the third: 10xy × 7z = 70xyz.

The section also discusses multiplying monomials with like bases, ensuring that students know to add the exponents when the bases are the same. Another example is multiplying 4xy × 5x²y² × 6x³y³. Here, we multiply the coefficients first: 4 × 5 × 6 = 120, and then the variables, combining like bases to yield: 120x^(1+2+3)y^(1+2+3) = 120x^6y^6.

This detailed exposition not only teaches the mechanics of multiplication but emphasizes their practical applications in geometry and volume calculations as well.

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Audio Book

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Introduction to Multiplying Monomials

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Observe the following examples.

(i) 2x × 5y × 7z = (2x × 5y) × 7z = 10xy × 7z = 70xyz

(ii) 4xy × 5x2y2 × 6x3y3 = (4xy × 5x2y2) × 6x3y3 = 20x3y3 × 6x3y3 = 120x3y3 × x3y3
= 120 (x3 × x3) × (y3 × y3) = 120x6 × y6.

Detailed Explanation

In this section, we show how to multiply multiple monomials together. The key is to group the first two monomials together and multiply them, then multiply this result by the next monomial. This process can be repeated for any number of monomials. For example:
1. Start with 2x, 5y, and 7z. First, multiply 2x with 5y to get 10xy, then multiply that result by 7z to get 70xyz.
2. In the second example, multiply 4xy by 5x2y2 first to get 20x3y3. Next, multiply that result by 6x3y3 to get 120x6y6. In both cases, we combine coefficients and apply the laws of exponents for the variables.

Examples & Analogies

Think of this process like cooking, where each ingredient represents a monomial. First, you mix two ingredients (monomials) together to create a batter (intermediate result), and then you add an additional ingredient to that batter to create your final dish (final result).

Step-by-Step Method

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It is clear that we first multiply the first two monomials and then multiply the resulting monomial by the third monomial. This method can be extended to the product of any number of monomials.

Detailed Explanation

The multiplication of three or more monomials can always be done in steps. You simply take the first two monomials, multiply them to get a new monomial, and then take that resulting monomial and multiply it by the next one. This pattern continues until all monomials are multiplied. The order in which you multiply does not matter, thanks to the associative property of multiplication, so you can adjust your steps based on which pairs may be easier to multiply first.

Examples & Analogies

Imagine building a complex structure with blocks. Each block represents a monomial. You can first connect two blocks together to form a bigger piece, and then attach another block to that larger piece. Regardless of which blocks you choose to combine first, you will eventually end up with the same larger structure.

Examples and Exercises

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TRY THESE We can find the product in other way also.

4xy × 5x2y2 × 6x3 y3
Find 4x × 5y × 7z
= (4 × 5 × 6) × (x × x2 × x3) × (y × y2 × y3)
First find 4x × 5y and multiply it by 7z; = 120 x6y6 or first find 5y × 7z and multiply it by 4x.
Is the result the same? What do you observe? Does the order in which you carry out the multiplication matter?

Detailed Explanation

In this chunk, we are given a practical example to apply the multiplication process we've learned. We take the expression 4xy × 5x2y2 × 6x3y3 and can break it down into its individual components. If we multiply the coefficients together (4 × 5 × 6) and then combine the variables by adding their exponents, we find that the final result is consistent regardless of the order: the rules of multiplication guarantee this consistency according to the associative property.

Examples & Analogies

This is similar to gathering a group of friends for a party. Whether you invite the first group of friends or the second group first, by the end of your planning, you will have the same number of friends over. You just have to connect them in different orders!

Definitions & Key Concepts

Learn essential terms and foundational ideas that form the basis of the topic.

Key Concepts

  • Multiplication of Monomials: The process of multiplying monomial coefficients and adding the exponents of like bases.

  • Associative Property: The order in which you multiply monomials does not matter.

  • Combining Like Terms: During multiplication, like variables should have their exponents added.

Examples & Real-Life Applications

See how the concepts apply in real-world scenarios to understand their practical implications.

Examples

  • Example of three monomials: 2x, 5y, and 7z gives 70xyz.

  • Example of complex monomials: 4xy × 5x^2y^2 × 6x^3y^3 results in 120x^6y^6.

Memory Aids

Use mnemonics, acronyms, or visual cues to help remember key information more easily.

🎵 Rhymes Time

  • When you multiply monomials, be wise, coefficients multiplied and exponents added, that's the prize!

📖 Fascinating Stories

  • Imagine a garden where each flower grows taller each year; the height doubles and your job is to mix flowers. As you mix, remember to add the height from previous years, this is how we multiply and grow in understanding!

🧠 Other Memory Gems

  • MICE: Multiply coefficients, Identify like terms, Combine exponents, and Evaluate the result.

🎯 Super Acronyms

M2E

  • Multiply
  • then Merge exponents!

Flash Cards

Review key concepts with flashcards.

Glossary of Terms

Review the Definitions for terms.

  • Term: Monomial

    Definition:

    An algebraic expression that consists of a single term.

  • Term: Coefficient

    Definition:

    The numerical factor in a term of an expression.

  • Term: Exponent

    Definition:

    The power to which a number or expression is raised.

  • Term: Like Bases

    Definition:

    Variables that are the same and whose exponents can be added.