Expand And Simplify The Expression: { (\sqrt{2}+8)(2 \sqrt{2}+3) - \sqrt{2}(6 \sqrt{2}+7)$}$
Introduction
Algebraic expressions are a fundamental concept in mathematics, and expanding and simplifying them is a crucial skill for students and professionals alike. In this article, we will explore the process of expanding and simplifying the given expression: {(\sqrt{2}+8)(2 \sqrt{2}+3) - \sqrt{2}(6 \sqrt{2}+7)$}$. We will break down the expression into manageable parts, apply the distributive property, and simplify the resulting expression.
Understanding the Expression
The given expression is a combination of two binomial expressions multiplied together and then subtracted by another binomial expression. To simplify this expression, we need to apply the distributive property, which states that for any real numbers a, b, and c:
a(b + c) = ab + ac
Expanding the First Binomial Expression
The first binomial expression is {\sqrt{2}+8}$, and we need to multiply it by ${2 \sqrt{2}+3}$.
{(\sqrt{2}+8)(2 \sqrt{2}+3) = \sqrt{2}(2 \sqrt{2}+3) + 8(2 \sqrt{2}+3)$
Using the distributive property, we can expand this expression as:
[$\sqrt{2}(2 \sqrt{2}+3) = 2(\sqrt{2})^2 + 3\sqrt{2}$
[$8(2 \sqrt{2}+3) = 16\sqrt{2} + 24$
Combining these two expressions, we get:
[$2(\sqrt{2})^2 + 3\sqrt{2} + 16\sqrt{2} + 24$
Simplifying this expression further, we get:
[$2(2) + 19\sqrt{2} + 24$
[$4 + 19\sqrt{2} + 24$
[$28 + 19\sqrt{2}$
Expanding the Second Binomial Expression
The second binomial expression is {\sqrt{2}(6 \sqrt{2}+7)}$, and we need to simplify it.
[$\sqrt{2}(6 \sqrt{2}+7) = 6(\sqrt{2})^2 + 7\sqrt{2}$
Using the distributive property, we can expand this expression as:
[$6(\sqrt{2})^2 = 6(2) = 12$
[$7\sqrt{2}$
Combining these two expressions, we get:
[$12 + 7\sqrt{2}$
Simplifying the Expression
Now that we have expanded both binomial expressions, we can simplify the original expression by subtracting the second expression from the first expression.
[$(\sqrt{2}+8)(2 \sqrt{2}+3) - \sqrt{2}(6 \sqrt{2}+7) = (28 + 19\sqrt{2}) - (12 + 7\sqrt{2})$
Using the distributive property, we can simplify this expression as:
[$28 + 19\sqrt{2} - 12 - 7\sqrt{2}$
Combining like terms, we get:
[$16 + 12\sqrt{2}$
Conclusion
In this article, we expanded and simplified the given expression: [(\sqrt{2}+8)(2 \sqrt{2}+3) - \sqrt{2}(6 \sqrt{2}+7)\$}. We applied the distributive property to expand both binomial expressions and then simplified the resulting expression by combining like terms. The final simplified expression is ${$16 + 12\sqrt{2}$.
Tips and Tricks
- When expanding and simplifying algebraic expressions, it's essential to apply the distributive property to each term.
- Use parentheses to group like terms and make it easier to simplify the expression.
- When combining like terms, make sure to add or subtract the coefficients of the same variables.
Common Mistakes
- Failing to apply the distributive property to each term.
- Not using parentheses to group like terms.
- Not combining like terms correctly.
Real-World Applications
Expanding and simplifying algebraic expressions is a crucial skill in various fields, including:
- Physics: To solve problems involving motion, energy, and momentum.
- Engineering: To design and optimize systems, such as bridges, buildings, and electronic circuits.
- Computer Science: To develop algorithms and data structures for solving complex problems.
Introduction
In our previous article, we explored the process of expanding and simplifying the given expression: [(\sqrt{2}+8)(2 \sqrt{2}+3) - \sqrt{2}(6 \sqrt{2}+7)\$}. We applied the distributive property, combined like terms, and arrived at the final simplified expression: [$16 + 12\sqrt{2}$.
In this article, we will address some common questions and concerns related to expanding and simplifying algebraic expressions. Whether you're a student, teacher, or professional, this Q&A guide will help you better understand the concepts and techniques involved.
Q: What is the distributive property, and how is it used in expanding algebraic expressions?
A: The distributive property is a fundamental concept in algebra that states that for any real numbers a, b, and c:
a(b + c) = ab + ac
This property is used to expand algebraic expressions by multiplying each term inside the parentheses by the term outside the parentheses.
Q: How do I apply the distributive property to expand an algebraic expression?
A: To apply the distributive property, follow these steps:
- Identify the terms inside the parentheses.
- Multiply each term inside the parentheses by the term outside the parentheses.
- Combine like terms to simplify the expression.
Q: What are like terms, and how do I combine them?
A: Like terms are terms that have the same variable raised to the same power. To combine like terms, add or subtract the coefficients of the same variables.
For example, in the expression [$2x + 3x + 4y - 2y$, the like terms are [$2x + 3x$] and [$4y - 2y$. Combining these like terms, we get:
[$5x + 2y$
Q: How do I simplify an algebraic expression?
A: To simplify an algebraic expression, follow these steps:
- Apply the distributive property to expand the expression.
- Combine like terms to simplify the expression.
- Remove any unnecessary parentheses or brackets.
Q: What are some common mistakes to avoid when expanding and simplifying algebraic expressions?
A: Some common mistakes to avoid include:
- Failing to apply the distributive property to each term.
- Not using parentheses to group like terms.
- Not combining like terms correctly.
- Not removing unnecessary parentheses or brackets.
Q: How do I apply the distributive property to expressions with multiple variables?
A: To apply the distributive property to expressions with multiple variables, follow these steps:
- Identify the terms inside the parentheses.
- Multiply each term inside the parentheses by the term outside the parentheses.
- Combine like terms to simplify the expression.
For example, in the expression [$(x + 2y)(3x + 4y)$, we can apply the distributive property as follows:
[$(x + 2y)(3x + 4y) = x(3x + 4y) + 2y(3x + 4y)$
[$= 3x^2 + 4xy + 6xy + 8y^2$
[$= 3x^2 + 10xy + 8y^2$
Q: How do I apply the distributive property to expressions with negative coefficients?
A: To apply the distributive property to expressions with negative coefficients, follow these steps:
- Identify the terms inside the parentheses.
- Multiply each term inside the parentheses by the term outside the parentheses.
- Combine like terms to simplify the expression.
For example, in the expression [$(-x + 2y)(3x + 4y)$, we can apply the distributive property as follows:
[$(-x + 2y)(3x + 4y) = -x(3x + 4y) + 2y(3x + 4y)$
[$= -3x^2 - 4xy + 6xy + 8y^2$
[$= -3x^2 + 2xy + 8y^2$
Conclusion
In this Q&A guide, we addressed some common questions and concerns related to expanding and simplifying algebraic expressions. By mastering the distributive property and combining like terms, you can tackle complex problems in various fields and make a significant impact in your chosen profession.
Tips and Tricks
- When expanding and simplifying algebraic expressions, it's essential to apply the distributive property to each term.
- Use parentheses to group like terms and make it easier to simplify the expression.
- When combining like terms, make sure to add or subtract the coefficients of the same variables.
Real-World Applications
Expanding and simplifying algebraic expressions is a crucial skill in various fields, including:
- Physics: To solve problems involving motion, energy, and momentum.
- Engineering: To design and optimize systems, such as bridges, buildings, and electronic circuits.
- Computer Science: To develop algorithms and data structures for solving complex problems.
By mastering the art of expanding and simplifying algebraic expressions, you can tackle complex problems in various fields and make a significant impact in your chosen profession.