Express The Product ( 2 + 5 ) ( 2 + 5 (\sqrt{2}+\sqrt{5})(\sqrt{2}+\sqrt{5} ( 2 + 5 ) ( 2 + 5 ] In Simplest Form.

Mar 10, 2025 by ADMIN 122 views

**Simplifying Radical Expressions: A Step-by-Step Guide**

Introduction

Radical expressions are a fundamental concept in algebra, and simplifying them is a crucial skill for any math enthusiast. In this article, we will explore how to simplify the product of two radical expressions, $(\sqrt{2}+\sqrt{5})(\sqrt{2}+\sqrt{5})$ . We will break down the process into manageable steps, making it easy to follow and understand.

Understanding Radical Expressions

Before we dive into simplifying the given expression, let's take a moment to understand what radical expressions are. A radical expression is any expression that contains a square root or a higher-order root. In this case, we have two radical expressions: $\sqrt{2}$ and $\sqrt{5}$ .

The Distributive Property

To simplify the given expression, we will use the distributive property, 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.

Expanding the Expression

Using the distributive property, we can expand the given expression as follows:

$(\sqrt{2}+\sqrt{5})(\sqrt{2}+\sqrt{5})$

$= (\sqrt{2})(\sqrt{2}) + (\sqrt{2})(\sqrt{5}) + (\sqrt{5})(\sqrt{2}) + (\sqrt{5})(\sqrt{5})$

$= 2 + \sqrt{10} + \sqrt{10} + 5$

Simplifying the Expression

Now that we have expanded the expression, we can simplify it by combining like terms. We can combine the two $\sqrt{10}$ terms into a single term, as they are like terms.

$= 2 + 2\sqrt{10} + 5$

$= 7 + 2\sqrt{10}$

Conclusion

In this article, we have simplified the product of two radical expressions, $(\sqrt{2}+\sqrt{5})(\sqrt{2}+\sqrt{5})$ . We used the distributive property to expand the expression and then simplified it by combining like terms. The final simplified expression is $7 + 2\sqrt{10}$ .

Tips and Tricks

When simplifying radical expressions, always look for like terms and combine them.
Use the distributive property to expand expressions and make them easier to simplify.
Be careful when combining like terms, as it's easy to make mistakes.

Common Mistakes to Avoid

Not using the distributive property to expand expressions.
Not combining like terms when simplifying expressions.
Making mistakes when combining like terms.

Real-World Applications

Simplifying radical expressions has many real-world applications, such as:

Calculating the area and perimeter of shapes with radical dimensions.
Solving equations with radical expressions.
Working with mathematical models that involve radical expressions.

Final Thoughts

Introduction

In our previous article, we explored how to simplify the product of two radical expressions, $(\sqrt{2}+\sqrt{5})(\sqrt{2}+\sqrt{5})$ . We used the distributive property to expand the expression and then simplified it by combining like terms. In this article, we will answer some common questions related to simplifying radical expressions.

Q&A

Q: What is the difference between a radical expression and a rational expression?

A: A radical expression is any expression that contains a square root or a higher-order root, while a rational expression is any expression that contains a fraction.

Q: How do I simplify a radical expression with multiple terms?

A: To simplify a radical expression with multiple terms, use the distributive property to expand the expression and then combine like terms.

Q: What is the order of operations when simplifying radical expressions?

A: The order of operations when simplifying radical expressions is:

Distribute the terms
Combine like terms
Simplify the expression

Q: Can I simplify a radical expression with a negative sign?

A: Yes, you can simplify a radical expression with a negative sign. To do this, use the distributive property to expand the expression and then combine like terms.

Q: How do I simplify a radical expression with a variable?

A: To simplify a radical expression with a variable, use the distributive property to expand the expression and then combine like terms.

Q: What is the difference between a simplified radical expression and a simplified rational expression?

A: A simplified radical expression is an expression that has been simplified by combining like terms and removing any unnecessary terms, while a simplified rational expression is an expression that has been simplified by canceling out any common factors.

Q: Can I simplify a radical expression with a decimal?

A: No, you cannot simplify a radical expression with a decimal. Radicals can only be simplified with integers or variables.

Q: How do I simplify a radical expression with a fraction?

A: To simplify a radical expression with a fraction, use the distributive property to expand the expression and then combine like terms.

Q: What is the difference between a simplified radical expression and a simplified expression with a square root?

A: A simplified radical expression is an expression that has been simplified by combining like terms and removing any unnecessary terms, while a simplified expression with a square root is an expression that has been simplified by removing any unnecessary terms.