(b+y) (by)= .......?

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Introduction

In algebra, expressions are a fundamental concept that helps us represent mathematical relationships between variables and constants. When we encounter expressions like (b+y) (by), it's essential to understand the rules of algebra that govern their simplification. In this article, we'll delve into the world of algebraic expressions, exploring the concept of the distributive property and how it applies to expressions like (b+y) (by).

Understanding 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. This property is denoted by the formula:

a(b+c) = ab + ac

where a, b, and c are variables or constants. The distributive property is a powerful tool that helps us simplify complex expressions and solve equations.

Applying the Distributive Property to (b+y) (by)

Now that we've understood the distributive property, let's apply it to the expression (b+y) (by). Using the distributive property, we can expand this expression as follows:

(b+y) (by) = b(by) + y(by)

Simplifying the Expression

To simplify the expression further, we can apply the associative property of multiplication, which states that:

(ab)c = a(bc)

Using this property, we can rewrite the expression as:

b(by) + y(by) = b(b) + b(y) + y(b) + y(y)

Combining Like Terms

Now that we've expanded the expression, we can combine like terms to simplify it further. Like terms are terms that have the same variable raised to the same power. In this case, we have:

b(b) + b(y) + y(b) + y(y) = b^2 + by + yb + y^2

Final Simplification

Using the commutative property of addition, which states that:

a + b = b + a

we can rewrite the expression as:

b^2 + by + yb + y^2 = b^2 + 2by + y^2

Conclusion

In this article, we've explored the concept of the distributive property and how it applies to expressions like (b+y) (by). We've seen how the distributive property allows us to expand expressions by multiplying each term inside the parentheses with the term outside. We've also applied the associative property of multiplication and combined like terms to simplify the expression. Finally, we've arrived at the simplified expression b^2 + 2by + y^2.

Frequently Asked Questions

  • What is the distributive property in algebra? 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.
  • How do I apply the distributive property to an expression? To apply the distributive property, simply multiply each term inside the parentheses with the term outside.
  • What is the associative property of multiplication? The associative property of multiplication states that (ab)c = a(bc), which means that the order in which we multiply numbers does not change the result.
  • What is the commutative property of addition? The commutative property of addition states that a + b = b + a, which means that the order in which we add numbers does not change the result.

Further Reading

  • Algebraic Expressions: A Comprehensive Guide
  • The Distributive Property: A Fundamental Concept in Algebra
  • Simplifying Algebraic Expressions: Tips and Tricks

References

  • [1] Algebra for Dummies, by Mary Jane Sterling
  • [2] Algebra: A Comprehensive Introduction, by Michael Artin
  • [3] The Art of Algebra, by Michael Artin

Note: The references provided are for informational purposes only and are not intended to be a comprehensive list of resources on the topic.

Q&A: Frequently Asked Questions about Algebraic Expressions

In this article, we'll address some of the most frequently asked questions about algebraic expressions, including the distributive property, associative property, and commutative property.

Q: What is the distributive property in algebra?

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 denoted by the formula:

a(b+c) = ab + ac

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

A: To apply the distributive property, simply multiply each term inside the parentheses with the term outside. For example, if we have the expression (b+y) (by), we can expand it as follows:

(b+y) (by) = b(by) + y(by)

Q: What is the associative property of multiplication?

A: The associative property of multiplication states that (ab)c = a(bc), which means that the order in which we multiply numbers does not change the result.

Q: What is the commutative property of addition?

A: The commutative property of addition states that a + b = b + a, which means that the order in which we add numbers does not change the result.

Q: How do I simplify algebraic expressions?

A: To simplify algebraic expressions, you can use the distributive property, associative property, and commutative property to expand and combine like terms.

Q: What are like terms in algebra?

A: Like terms are terms that have the same variable raised to the same power. For example, 2x and 3x are like terms because they both have the variable x raised to the power of 1.

Q: How do I combine like terms?

A: To combine like terms, simply add or subtract the coefficients of the like terms. For example, if we have the expression 2x + 3x, we can combine the like terms as follows:

2x + 3x = 5x

Q: What is the order of operations in algebra?

A: The order of operations in algebra is:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I solve algebraic equations?

A: To solve algebraic equations, you can use a variety of techniques, including:

  1. Adding or subtracting the same value to both sides of the equation.
  2. Multiplying or dividing both sides of the equation by the same value.
  3. Using inverse operations to isolate the variable.

Q: What are some common algebraic expressions?

A: Some common algebraic expressions include:

  1. Linear expressions: Expressions of the form ax + b, where a and b are constants.
  2. Quadratic expressions: Expressions of the form ax^2 + bx + c, where a, b, and c are constants.
  3. Polynomial expressions: Expressions of the form a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0, where a_n, a_(n-1), ..., a_1, and a_0 are constants.

Conclusion

In this article, we've addressed some of the most frequently asked questions about algebraic expressions, including the distributive property, associative property, and commutative property. We've also covered some common algebraic expressions and techniques for solving algebraic equations. Whether you're a student or a teacher, we hope this article has been helpful in your understanding of algebraic expressions.

Further Reading

  • Algebraic Expressions: A Comprehensive Guide
  • The Distributive Property: A Fundamental Concept in Algebra
  • Simplifying Algebraic Expressions: Tips and Tricks

References

  • [1] Algebra for Dummies, by Mary Jane Sterling
  • [2] Algebra: A Comprehensive Introduction, by Michael Artin
  • [3] The Art of Algebra, by Michael Artin