What Is The Simplest Form Of 2 2 3 − 2 \frac{2 \sqrt{2}}{\sqrt{3}-\sqrt{2}} 3 − 2 2 2 ?A. 2 6 + 4 2 \sqrt{6}+4 2 6 + 4 B. 2 5 + 4 2 \sqrt{5}+4 2 5 + 4 C. 2 6 + 4 5 \frac{2 \sqrt{6}+4}{5} 5 2 6 + 4 D. 2 5 + 4 5 \frac{2 \sqrt{5}+4}{5} 5 2 5 + 4

Mar 8, 2025 by ADMIN 263 views

$What is the simplest form of $\frac{2 \sqrt{2}}{\sqrt{3}-\sqrt{2}}$?$

Introduction

Rationalizing the denominator is a common technique used in algebra to simplify complex fractions. This technique involves multiplying the numerator and denominator by a specific value to eliminate any radicals in the denominator. In this article, we will explore how to rationalize the denominator of the given expression $\frac{2 \sqrt{2}}{\sqrt{3}-\sqrt{2}}$ and find its simplest form.

Understanding the Problem

The given expression is a fraction with a radical in the denominator. To simplify this expression, we need to rationalize the denominator, which means removing the radical from the denominator. This can be achieved by multiplying the numerator and denominator by a value that will eliminate the radical in the denominator.

Rationalizing the Denominator

To rationalize the denominator, we need to multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a binomial expression $a-b$ is $a+b$ . In this case, the conjugate of $\sqrt{3}-\sqrt{2}$ is $\sqrt{3}+\sqrt{2}$ .

import sympy as sp
x = sp.symbols('x')

expr = (2*sp.sqrt(2))/((sp.sqrt(3)-sp.sqrt(2)))

rationalized_expr = expr * ((sp.sqrt(3)+sp.sqrt(2))/(sp.sqrt(3)+sp.sqrt(2)))

simplified_expr = sp.simplify(rationalized_expr)

Simplifying the Expression

After rationalizing the denominator, we need to simplify the expression. This involves combining like terms and eliminating any unnecessary factors.

# Simplify the expression
simplified_expr = sp.simplify(simplified_expr)

Finding the Simplest Form

The simplified expression is the simplest form of the given expression. This is the final answer to the problem.

Conclusion

In this article, we explored how to rationalize the denominator of the given expression $\frac{2 \sqrt{2}}{\sqrt{3}-\sqrt{2}}$ and find its simplest form. We used the technique of multiplying the numerator and denominator by the conjugate of the denominator to eliminate the radical in the denominator. We then simplified the expression to find the simplest form.

Final Answer

The final answer to the problem is $\boxed{\frac{2 \sqrt{6}+4}{5}}$ .

Discussion

The given expression is a complex fraction with a radical in the denominator. To simplify this expression, we need to rationalize the denominator, which involves multiplying the numerator and denominator by a value that will eliminate the radical in the denominator. This technique is commonly used in algebra to simplify complex fractions.

Step-by-Step Solution

Here is a step-by-step solution to the problem:

Multiply the numerator and denominator by the conjugate of the denominator.
Simplify the expression by combining like terms and eliminating any unnecessary factors.
The simplified expression is the simplest form of the given expression.

Common Mistakes

Here are some common mistakes to avoid when rationalizing the denominator:

Not multiplying the numerator and denominator by the conjugate of the denominator.
Not simplifying the expression after rationalizing the denominator.
Not eliminating any unnecessary factors in the expression.

Real-World Applications

Rationalizing the denominator is a common technique used in algebra to simplify complex fractions. This technique has many real-world applications, including:

Simplifying complex expressions in physics and engineering.
Solving equations in calculus and differential equations.
Simplifying expressions in computer science and programming.

Conclusion

In conclusion, rationalizing the denominator is a powerful technique used in algebra to simplify complex fractions. By multiplying the numerator and denominator by the conjugate of the denominator, we can eliminate the radical in the denominator and simplify the expression. This technique has many real-world applications and is an essential tool for anyone working with complex fractions.

Introduction

Rationalizing the denominator is a common technique used in algebra to simplify complex fractions. In our previous article, we explored how to rationalize the denominator of the given expression $\frac{2 \sqrt{2}}{\sqrt{3}-\sqrt{2}}$ and find its simplest form. In this article, we will answer some frequently asked questions about rationalizing the denominator.

Q: What is rationalizing the denominator?

A: Rationalizing the denominator is a technique used in algebra to simplify complex fractions by eliminating any radicals in the denominator. This is achieved by multiplying the numerator and denominator by a value that will eliminate the radical in the denominator.

Q: Why do we need to rationalize the denominator?

A: We need to rationalize the denominator to simplify complex fractions and make them easier to work with. Rationalizing the denominator helps to eliminate any radicals in the denominator, making it easier to perform operations such as addition and subtraction.

Q: How do we rationalize the denominator?

A: To rationalize the denominator, we need to multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a binomial expression $a-b$ is $a+b$ . For example, if the denominator is $\sqrt{3}-\sqrt{2}$ , the conjugate is $\sqrt{3}+\sqrt{2}$ .

Q: What is the conjugate of a binomial expression?

A: The conjugate of a binomial expression $a-b$ is $a+b$ . For example, the conjugate of $\sqrt{3}-\sqrt{2}$ is $\sqrt{3}+\sqrt{2}$ .

Q: How do we simplify the expression after rationalizing the denominator?

A: After rationalizing the denominator, we need to simplify the expression by combining like terms and eliminating any unnecessary factors.

Q: What are some common mistakes to avoid when rationalizing the denominator?

A: Some common mistakes to avoid when rationalizing the denominator include:

Not multiplying the numerator and denominator by the conjugate of the denominator.
Not simplifying the expression after rationalizing the denominator.
Not eliminating any unnecessary factors in the expression.

Q: What are some real-world applications of rationalizing the denominator?

A: Rationalizing the denominator has many real-world applications, including:

Simplifying complex expressions in physics and engineering.
Solving equations in calculus and differential equations.
Simplifying expressions in computer science and programming.

Q: Can you provide an example of rationalizing the denominator?

A: Here is an example of rationalizing the denominator:

$\frac{2 \sqrt{2}}{\sqrt{3}-\sqrt{2}}$

To rationalize the denominator, we need to multiply the numerator and denominator by the conjugate of the denominator, which is $\sqrt{3}+\sqrt{2}$ .

$\frac{2 \sqrt{2}}{\sqrt{3}-\sqrt{2}} \cdot \frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}+\sqrt{2}}$

Simplifying the expression, we get:

$\frac{2 \sqrt{6}+4}{5}$

Q: How do we know when to rationalize the denominator?

A: We need to rationalize the denominator when the denominator contains a radical. This is because radicals can make it difficult to perform operations such as addition and subtraction.

Q: Can you provide a step-by-step solution to rationalizing the denominator?

A: Here is a step-by-step solution to rationalizing the denominator:

Multiply the numerator and denominator by the conjugate of the denominator.
Simplify the expression by combining like terms and eliminating any unnecessary factors.
The simplified expression is the simplest form of the given expression.