Expand And Fully Simplify 7 ( 7 + 2 \sqrt{7}(\sqrt{7}+\sqrt{2} 7 ( 7 + 2 ].

Mar 9, 2025 by ADMIN 82 views

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

Introduction

Radical expressions are a fundamental concept in mathematics, and understanding how to expand and simplify them is crucial for solving various mathematical problems. In this article, we will focus on expanding and simplifying the radical expression $\sqrt{7}(\sqrt{7}+\sqrt{2})$ . We will break down the process into manageable steps, making it easier to understand and apply.

Understanding Radical Expressions

Before we dive into the expansion and simplification process, let's briefly review what radical expressions are. A radical expression is a mathematical expression that contains a square root or other roots. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16.

Expanding the Radical Expression

To expand the radical expression $\sqrt{7}(\sqrt{7}+\sqrt{2})$ , we need to apply the distributive property, which states that for any real numbers a, b, and c, a(b + c) = ab + ac. In this case, we can distribute the $\sqrt{7}$ to both terms inside the parentheses.

$\sqrt{7}(\sqrt{7}+\sqrt{2}) = \sqrt{7}\cdot\sqrt{7} + \sqrt{7}\cdot\sqrt{2}$

Simplifying the Expression

Now that we have expanded the expression, let's simplify it by combining like terms. The first term, $\sqrt{7}\cdot\sqrt{7}$ , can be simplified by multiplying the two square roots together.

$\sqrt{7}\cdot\sqrt{7} = \sqrt{7^2} = 7$

The second term, $\sqrt{7}\cdot\sqrt{2}$ , cannot be simplified further, as the two square roots are different.

$\sqrt{7}\cdot\sqrt{2} = \sqrt{7\cdot2} = \sqrt{14}$

Combining Like Terms

Now that we have simplified the two terms, we can combine them by adding them together.

$7 + \sqrt{14}$

Conclusion

In this article, we expanded and simplified the radical expression $\sqrt{7}(\sqrt{7}+\sqrt{2})$ . We applied the distributive property to expand the expression and then simplified it by combining like terms. The final simplified expression is $7 + \sqrt{14}$ .

Real-World Applications

Radical expressions have numerous real-world applications, including:

Physics: Radical expressions are used to describe the motion of objects, such as the trajectory of a projectile.
Engineering: Radical expressions are used to calculate the stress and strain on materials, such as the tension on a rope.
Computer Science: Radical expressions are used in algorithms for solving mathematical problems, such as the quadratic formula.

Tips and Tricks

Here are some tips and tricks for expanding and simplifying radical expressions:

Use the distributive property: The distributive property is a powerful tool for expanding radical expressions.
Simplify like terms: Combining like terms is a crucial step in simplifying radical expressions.
Use the order of operations: The order of operations (PEMDAS) is a mnemonic device that helps you remember the order in which to perform mathematical operations.

Common Mistakes

Here are some common mistakes to avoid when expanding and simplifying radical expressions:

Forgetting to distribute: Failing to distribute the radical expression to both terms inside the parentheses.
Not simplifying like terms: Failing to combine like terms can lead to incorrect answers.
Using the wrong order of operations: Using the wrong order of operations can lead to incorrect answers.

Conclusion

Q: What is the distributive property, and how is it used in expanding radical expressions?

A: The distributive property is a mathematical concept that states that for any real numbers a, b, and c, a(b + c) = ab + ac. In the context of expanding radical expressions, the distributive property is used to distribute the radical expression to both terms inside the parentheses. For example, in the expression $\sqrt{7}(\sqrt{7}+\sqrt{2})$ , the distributive property is used to expand it as $\sqrt{7}\cdot\sqrt{7} + \sqrt{7}\cdot\sqrt{2}$ .

Q: How do I simplify like terms in a radical expression?

A: To simplify like terms in a radical expression, you need to combine the terms that have the same radical. For example, in the expression $3\sqrt{2} + 2\sqrt{2}$ , the two terms have the same radical, so you can combine them as $5\sqrt{2}$ .

Q: What is the order of operations, and how is it used in simplifying radical expressions?

A: The order of operations is a mnemonic device that helps you remember the order in which to perform mathematical operations. The order of operations is:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate exponents next.
Multiplication and Division: Evaluate multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate addition and subtraction operations from left to right.

In the context of simplifying radical expressions, the order of operations is used to ensure that the correct operations are performed in the correct order.

Q: How do I handle negative numbers in radical expressions?

A: When a negative number is present in a radical expression, you need to handle it carefully. The square root of a negative number is an imaginary number, which is denoted by the letter i. For example, the square root of -1 is denoted as i. When simplifying radical expressions with negative numbers, you need to consider the imaginary unit i.

Q: Can I simplify radical expressions with variables?

A: Yes, you can simplify radical expressions with variables. When simplifying radical expressions with variables, you need to consider the properties of radicals and the rules for simplifying expressions with variables.

Q: How do I know when to use the conjugate to simplify a radical expression?

A: The conjugate is a mathematical concept that is used to simplify radical expressions. The conjugate of a binomial expression is obtained by changing the sign of the second term. For example, the conjugate of $a + b$ is $a - b$ . When simplifying radical expressions, you can use the conjugate to eliminate the radical.

Q: Can I use a calculator to simplify radical expressions?

A: Yes, you can use a calculator to simplify radical expressions. However, it's always a good idea to check your work by hand to ensure that the calculator is giving you the correct answer.

Q: How do I know when to use the rationalizing factor to simplify a radical expression?

A: The rationalizing factor is a mathematical concept that is used to simplify radical expressions. The rationalizing factor is a number that, when multiplied by the radical expression, eliminates the radical. When simplifying radical expressions, you can use the rationalizing factor to eliminate the radical.

Conclusion

In conclusion, expanding and simplifying radical expressions is a crucial skill for solving mathematical problems. By understanding the distributive property, simplifying like terms, and using the order of operations, you can simplify radical expressions with ease. Remember to use the tips and tricks provided in this article to avoid common mistakes and achieve accurate results.