Write ( 4 − 5 ) ( 4 + 5 ) 2 11 \frac{(4-\sqrt{5})(4+\sqrt{5})}{2 \sqrt{11}} 2 11 ​ ( 4 − 5 ​ ) ( 4 + 5 ​ ) ​ In The Form A B \frac{\sqrt{a}}{b} B A ​ ​ , Where A A A And B B B Are Integers.Find The Values Of A A A And B B B .

Introduction

When dealing with expressions that involve square roots in the denominator, it's essential to rationalize the denominator to simplify the expression. Rationalizing the denominator involves multiplying the numerator and denominator by a specific value to eliminate the square root from the denominator. In this article, we will explore how to rationalize the denominator of the given expression $\frac{(4-\sqrt{5})(4+\sqrt{5})}{2 \sqrt{11}}$ and simplify it to the form $\frac{\sqrt{a}}{b}$, where $a$ and $b$ are integers.

Understanding the Concept of Rationalizing the Denominator

Rationalizing the denominator is a process used to eliminate square roots from the denominator of a fraction. This is achieved by multiplying the numerator and denominator by a specific value that will eliminate the square root. The process involves using the conjugate of the denominator to rationalize it. The conjugate of a binomial expression $a + b$ is $a - b$, and vice versa.

Rationalizing the Denominator of the Given Expression

To rationalize the denominator of the given expression $\frac{(4-\sqrt{5})(4+\sqrt{5})}{2 \sqrt{11}}$, we need to multiply the numerator and denominator by the conjugate of the denominator, which is $2 \sqrt{11}$. This will eliminate the square root from the denominator.

Step 1: Multiply the Numerator and Denominator by the Conjugate of the Denominator

To rationalize the denominator, we multiply the numerator and denominator by $2 \sqrt{11}$.

$\frac{(4-\sqrt{5})(4+\sqrt{5})}{2 \sqrt{11}} \cdot \frac{2 \sqrt{11}}{2 \sqrt{11}}$

Step 2: Simplify the Expression

Now, we simplify the expression by multiplying the numerator and denominator.

$\frac{(4-\sqrt{5})(4+\sqrt{5}) \cdot 2 \sqrt{11}}{2 \sqrt{11} \cdot 2 \sqrt{11}}$

Step 3: Expand and Simplify the Numerator

Next, we expand and simplify the numerator.

$(4-\sqrt{5})(4+\sqrt{5}) = 16 - 5 = 11$

So, the expression becomes:

$\frac{11 \cdot 2 \sqrt{11}}{2 \sqrt{11} \cdot 2 \sqrt{11}}$

Step 4: Simplify the Expression

Now, we simplify the expression by canceling out the common factors.

$\frac{11 \cdot 2 \sqrt{11}}{2 \sqrt{11} \cdot 2 \sqrt{11}} = \frac{11}{2 \sqrt{11}} \cdot \frac{2 \sqrt{11}}{2 \sqrt{11}}$

Step 5: Simplify the Expression

Finally, we simplify the expression by canceling out the common factors.

$\frac{11}{2 \sqrt{11}} \cdot \frac{2 \sqrt{11}}{2 \sqrt{11}} = \frac{11 \cdot 2 \sqrt{11}}{2 \sqrt{11} \cdot 2 \sqrt{11}} = \frac{11 \cdot 2 \sqrt{11}}{44}$

Step 6: Simplify the Expression

Now, we simplify the expression by canceling out the common factors.

$\frac{11 \cdot 2 \sqrt{11}}{44} = \frac{22 \sqrt{11}}{44}$

Step 7: Simplify the Expression

Finally, we simplify the expression by canceling out the common factors.

$\frac{22 \sqrt{11}}{44} = \frac{\sqrt{11}}{2}$

Conclusion

In this article, we have learned how to rationalize the denominator of the given expression $\frac{(4-\sqrt{5})(4+\sqrt{5})}{2 \sqrt{11}}$ and simplify it to the form $\frac{\sqrt{a}}{b}$, where $a$ and $b$ are integers. We have used the concept of rationalizing the denominator to eliminate the square root from the denominator and simplify the expression. The final simplified expression is $\frac{\sqrt{11}}{2}$.

Values of $a$ and $b$

The values of $a$ and $b$ are:

$a = 11$ $b = 2$

Final Answer

The final answer is $\boxed{\frac{\sqrt{11}}{2}}$.

Introduction

In our previous article, we explored how to rationalize the denominator of the given expression $\frac{(4-\sqrt{5})(4+\sqrt{5})}{2 \sqrt{11}}$ and simplify it to the form $\frac{\sqrt{a}}{b}$, where $a$ and $b$ are integers. In this article, we will answer some frequently asked questions related to rationalizing the denominator and provide additional examples to help you understand the concept better.

Q: What is rationalizing the denominator?

A: Rationalizing the denominator is a process used to eliminate square roots from the denominator of a fraction. This is achieved by multiplying the numerator and denominator by a specific value that will eliminate the square root.

Q: Why do we need to rationalize the denominator?

A: We need to rationalize the denominator to simplify the expression and make it easier to work with. Rationalizing the denominator eliminates the square root from 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 multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a binomial expression $a + b$ is $a - b$, and vice versa.

Q: What is the conjugate of a binomial expression?

A: The conjugate of a binomial expression $a + b$ is $a - b$, and vice versa.

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

A: After rationalizing the denominator, we simplify the expression by canceling out the common factors.

Q: Can we rationalize the denominator of any expression?

A: No, we can only rationalize the denominator of expressions that have a square root in the denominator.

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 canceling out the common factors

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

A: We know when to rationalize the denominator when we have an expression with a square root in the denominator and we need to simplify it.

Q: Can we rationalize the denominator of a fraction with a negative exponent?

A: Yes, we can rationalize the denominator of a fraction with a negative exponent by multiplying the numerator and denominator by the conjugate of the denominator.

Q: How do we rationalize the denominator of a fraction with a negative exponent?

A: To rationalize the denominator of a fraction with a negative exponent, we multiply the numerator and denominator by the conjugate of the denominator.

Q: Can we rationalize the denominator of a fraction with a complex number in the denominator?

A: Yes, we can rationalize the denominator of a fraction with a complex number in the denominator by multiplying the numerator and denominator by the conjugate of the denominator.

Q: How do we rationalize the denominator of a fraction with a complex number in the denominator?

A: To rationalize the denominator of a fraction with a complex number in the denominator, we multiply the numerator and denominator by the conjugate of the denominator.

Conclusion

In this article, we have answered some frequently asked questions related to rationalizing the denominator and provided additional examples to help you understand the concept better. We have also discussed some common mistakes to avoid when rationalizing the denominator and how to know when to rationalize the denominator.

Final Answer

The final answer is $\boxed{\frac{\sqrt{11}}{2}}$.