Simplify The Expression: \[$(\sqrt{2} + 8)(\sqrt{2} + 3)\$\]
Introduction
In this article, we will simplify the given expression {(\sqrt{2} + 8)(\sqrt{2} + 3)$}$. This involves using the distributive property of multiplication over addition, which states that for any real numbers a, b, and c, a(b + c) = ab + ac. We will apply this property to expand the given expression and simplify it to its final form.
Step 1: Apply the Distributive Property
To simplify the expression, we will first apply the distributive property to expand the product of the two binomials. This means that we will multiply each term in the first binomial by each term in the second binomial.
{(\sqrt{2} + 8)(\sqrt{2} + 3) = (\sqrt{2})(\sqrt{2}) + (\sqrt{2})(3) + (8)(\sqrt{2}) + (8)(3)$}$
Step 2: Simplify the Terms
Now that we have expanded the expression, we can simplify each term. The first term is the product of and , which is equal to 2. The second term is the product of and 3, which is equal to . The third term is the product of 8 and , which is equal to . The fourth term is the product of 8 and 3, which is equal to 24.
{= 2 + 3\sqrt{2} + 8\sqrt{2} + 24$}$
Step 3: Combine Like Terms
Now that we have simplified each term, we can combine like terms. The terms and are like terms, so we can combine them by adding their coefficients. This gives us .
{= 2 + 11\sqrt{2} + 24$}$
Step 4: Simplify the Expression
Finally, we can simplify the expression by combining the constant terms. The constant terms are 2 and 24, so we can combine them by adding them. This gives us 26.
{= 26 + 11\sqrt{2}$}$
Conclusion
In this article, we simplified the expression {(\sqrt{2} + 8)(\sqrt{2} + 3)$}$ using the distributive property and combining like terms. We expanded the expression, simplified each term, and combined like terms to arrive at the final simplified form of the expression.
Final Answer
The final simplified form of the expression {(\sqrt{2} + 8)(\sqrt{2} + 3)$}$ is ${26 + 11\sqrt{2}\$}.
Why is this Important?
Simplifying expressions is an important skill in mathematics, as it allows us to manipulate and solve equations more easily. In this article, we used the distributive property and combining like terms to simplify the expression {(\sqrt{2} + 8)(\sqrt{2} + 3)$}$. This is an important skill to have in mathematics, as it can be used to solve a wide range of problems.
Real-World Applications
Simplifying expressions has many real-world applications. For example, in physics, we often need to simplify complex expressions in order to solve problems. In engineering, we use simplifying expressions to design and build complex systems. In finance, we use simplifying expressions to calculate interest rates and investment returns.
Common Mistakes
When simplifying expressions, there are several common mistakes that we can make. One common mistake is to forget to combine like terms. Another common mistake is to simplify the expression incorrectly. To avoid these mistakes, it is essential to carefully read and follow the instructions, and to double-check our work.
Tips and Tricks
When simplifying expressions, there are several tips and tricks that we can use to make the process easier. One tip is to use the distributive property to expand the expression. Another tip is to combine like terms as soon as possible. Finally, it is essential to double-check our work to ensure that we have simplified the expression correctly.
Conclusion
Q: What is the distributive property?
A: The distributive property is a mathematical concept that states that for any real numbers a, b, and c, a(b + c) = ab + ac. This means that we can multiply a single value by each term in a binomial.
Q: How do I apply the distributive property to simplify an expression?
A: To apply the distributive property, we need to multiply each term in the first binomial by each term in the second binomial. This will give us a new expression with multiple terms.
Q: What is the difference between like terms and unlike terms?
A: Like terms are terms that have the same variable and exponent. Unlike terms are terms that have different variables or exponents.
Q: How do I combine like terms?
A: To combine like terms, we need to add or subtract the coefficients of the like terms. For example, if we have 2x + 3x, we can combine the like terms by adding the coefficients: 2x + 3x = 5x.
Q: What is the final simplified form of the expression {(\sqrt{2} + 8)(\sqrt{2} + 3)$}$?
A: The final simplified form of the expression {(\sqrt{2} + 8)(\sqrt{2} + 3)$}$ is ${26 + 11\sqrt{2}\$}.
Q: Why is it important to simplify expressions?
A: Simplifying expressions is important because it allows us to manipulate and solve equations more easily. It also helps us to identify patterns and relationships between variables.
Q: What are some common mistakes to avoid when simplifying expressions?
A: Some common mistakes to avoid when simplifying expressions include forgetting to combine like terms, simplifying the expression incorrectly, and not double-checking our work.
Q: What are some tips and tricks for simplifying expressions?
A: Some tips and tricks for simplifying expressions include using the distributive property to expand the expression, combining like terms as soon as possible, and double-checking our work to ensure that we have simplified the expression correctly.
Q: How can I apply the concepts learned in this article to real-world problems?
A: The concepts learned in this article can be applied to a wide range of real-world problems, including physics, engineering, and finance. For example, in physics, we can use simplifying expressions to solve problems involving motion and energy. In engineering, we can use simplifying expressions to design and build complex systems. In finance, we can use simplifying expressions to calculate interest rates and investment returns.
Q: What are some additional resources for learning more about simplifying expressions?
A: Some additional resources for learning more about simplifying expressions include textbooks, online tutorials, and practice problems. You can also try searching for videos and articles on the topic to get more practice and review.
Conclusion
In this article, we answered some common questions about simplifying expressions, including how to apply the distributive property, how to combine like terms, and how to avoid common mistakes. We also provided some tips and tricks for simplifying expressions and discussed how to apply the concepts learned in this article to real-world problems. By following the tips and tricks outlined in this article, you can simplify expressions more easily and accurately.