(7xy+4z)×(3xy×2z) Answer The Question

Understanding the Problem

When dealing with algebraic expressions, multiplication is a fundamental operation that requires careful application of the rules of exponents and the distributive property. In this article, we will delve into the process of multiplying two algebraic expressions, specifically (7xy+4z) and (3xy×2z), to arrive at the final result.

The Distributive Property: A Key Concept

The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses with the term outside. In the context of our problem, we will apply the distributive property to multiply each term in (7xy+4z) with each term in (3xy×2z).

Step 1: Multiply Each Term in (7xy+4z) with Each Term in (3xy×2z)

To begin, we will multiply each term in (7xy+4z) with each term in (3xy×2z). This will involve multiplying each term in the first expression with each term in the second expression, and then combining like terms.

Multiplying (7xy) with (3xy×2z)

First, we will multiply (7xy) with (3xy×2z). Using the distributive property, we can rewrite this as:

(7xy) × (3xy) × (2z)

Applying the Rules of Exponents

When multiplying variables with the same base, we add their exponents. In this case, we have:

(7xy) × (3xy) = 21x^2y2

Multiplying by (2z)

Next, we will multiply 21x^2y2 by (2z):

21x^2y2 × (2z) = 42x^2y2z

Multiplying (4z) with (3xy×2z)

Now, we will multiply (4z) with (3xy×2z). Using the distributive property, we can rewrite this as:

(4z) × (3xy) × (2z)

Applying the Rules of Exponents

When multiplying variables with the same base, we add their exponents. In this case, we have:

(4z) × (3xy) = 12xyz

Multiplying by (2z)

Next, we will multiply 12xyz by (2z):

12xyz × (2z) = 24xyz^2

Combining Like Terms

Now that we have multiplied each term in (7xy+4z) with each term in (3xy×2z), we can combine like terms to arrive at the final result.

The Final Result

Combining the results of the previous steps, we have:

42x^2y2z + 24xyz^2

Simplifying the Expression

We can simplify the expression by factoring out common terms. In this case, we can factor out 12x^2y2z:

12x^2y2z(3 + 2z/x)

Conclusion

In this article, we have walked through the process of multiplying two algebraic expressions, specifically (7xy+4z) and (3xy×2z), to arrive at the final result. By applying the distributive property and the rules of exponents, we have been able to simplify the expression and arrive at a final result of 12x^2y2z(3 + 2z/x). This demonstrates the importance of careful application of algebraic rules in solving complex problems.

Common Mistakes to Avoid

When multiplying algebraic expressions, it is easy to make mistakes. Here are some common mistakes to avoid:

• Not applying the distributive property: Failing to apply the distributive property can lead to incorrect results.
• Not following the rules of exponents: Failing to follow the rules of exponents can lead to incorrect results.
• Not combining like terms: Failing to combine like terms can lead to incorrect results.

Tips for Solving Algebraic Expressions

Here are some tips for solving algebraic expressions:

• Read the problem carefully: Before starting to solve the problem, read it carefully to understand what is being asked.
• Apply the distributive property: The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses with the term outside.
• Follow the rules of exponents: When multiplying variables with the same base, we add their exponents.
• Combine like terms: Combining like terms is an important step in simplifying algebraic expressions.

Real-World Applications

Algebraic expressions have many real-world applications. Here are a few examples:

• Physics: Algebraic expressions are used to describe the motion of objects in physics.
• Engineering: Algebraic expressions are used to design and optimize systems in engineering.
• Computer Science: Algebraic expressions are used to describe algorithms and data structures in computer science.

Conclusion

In conclusion, multiplying algebraic expressions is a fundamental concept in algebra that requires careful application of the rules of exponents and the distributive property. By following the steps outlined in this article, we can simplify complex expressions and arrive at a final result. Whether you are a student or a professional, understanding algebraic expressions is essential for solving complex problems in a variety of fields.

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses with the term outside. It is often represented by the equation a(b + c) = ab + ac.

Q: How do I apply the distributive property when multiplying algebraic expressions?

A: To apply the distributive property, you need to multiply each term inside the parentheses with the term outside. For example, if you have the expression (2x + 3)(x + 4), you would multiply 2x by x and 3 by x, and then multiply 2x by 4 and 3 by 4.

Q: What are the rules of exponents?

A: The rules of exponents are a set of rules that govern how to multiply and divide variables with the same base. When multiplying variables with the same base, you add their exponents. For example, if you have the expression x^2 × x^3, you would add the exponents to get x^(2+3) = x^5.

Q: How do I combine like terms when multiplying algebraic expressions?

A: To combine like terms, you need to identify the terms that have the same variable and exponent, and then add or subtract their coefficients. For example, if you have the expression 2x + 3x, you would combine the like terms to get 5x.

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

A: Some common mistakes to avoid when multiplying algebraic expressions include:

• Not applying the distributive property
• Not following the rules of exponents
• Not combining like terms
• Not simplifying the expression

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you need to combine like terms, eliminate any unnecessary parentheses, and rewrite the expression in a more compact form. For example, if you have the expression 2x + 3x + 4, you would combine the like terms to get 5x + 4.

Q: What are some real-world applications of multiplying algebraic expressions?

A: Some real-world applications of multiplying algebraic expressions include:

• Physics: Algebraic expressions are used to describe the motion of objects in physics.
• Engineering: Algebraic expressions are used to design and optimize systems in engineering.
• Computer Science: Algebraic expressions are used to describe algorithms and data structures in computer science.

Q: How do I use algebraic expressions in real-world problems?

A: To use algebraic expressions in real-world problems, you need to identify the variables and constants in the problem, and then use algebraic expressions to model the situation. For example, if you are designing a bridge, you might use algebraic expressions to model the stress and strain on the bridge.

Q: What are some tips for solving algebraic expressions?

A: Some tips for solving algebraic expressions include:

• Read the problem carefully
• Apply the distributive property
• Follow the rules of exponents
• Combine like terms
• Simplify the expression

Q: How do I check my work when solving algebraic expressions?

A: To check your work when solving algebraic expressions, you need to plug in values for the variables and constants, and then check if the expression is true. For example, if you have the expression 2x + 3x + 4, you could plug in x = 1 to get 2(1) + 3(1) + 4 = 9.

Q: What are some common algebraic expressions that I should know?

A: Some common algebraic expressions that you should know include:

• Linear expressions: ax + b
• Quadratic expressions: ax^2 + bx + c
• Polynomial expressions: a_n x^n + a_(n-1) x^(n-1) + ... + a_0

Q: How do I use algebraic expressions to solve systems of equations?

A: To use algebraic expressions to solve systems of equations, you need to use algebraic expressions to model the situation, and then use algebraic techniques to solve the system of equations. For example, if you have the system of equations x + y = 2 and x - y = 1, you could use algebraic expressions to model the situation and then solve the system of equations.

Q: What are some advanced algebraic expressions that I should know?

A: Some advanced algebraic expressions that you should know include:

• Rational expressions: a/b
• Radical expressions: sqrt(a)
• Exponential expressions: a^x

Q: How do I use algebraic expressions to solve optimization problems?

A: To use algebraic expressions to solve optimization problems, you need to use algebraic expressions to model the situation, and then use algebraic techniques to find the maximum or minimum value of the expression. For example, if you have the expression x^2 + 2x + 1, you could use algebraic expressions to model the situation and then find the maximum or minimum value of the expression.