The Conjugate of the Denominator

Oct 13, 2025 by windailys.com 28 views

Introduction

Rationalising the denominator is a crucial step in simplifying complex fractions, especially when dealing with square roots. In this problem, we are required to rationalise the expression 6-4√3 divided by 6+4√3. This involves eliminating the square root from the denominator and simplifying the resulting expression.

Understanding the Problem

To begin, let's examine the given expression: (6-4√3) / (6+4√3). Our goal is to rationalise the denominator, which means we need to eliminate the square root from the denominator. This can be achieved by multiplying both the numerator and the denominator by the conjugate of the denominator.

The Conjugate of the Denominator

The conjugate of a binomial expression is obtained by changing the sign of the second term. In this case, the conjugate of 6+4√3 is 6-4√3. By multiplying both the numerator and the denominator by the conjugate, we can eliminate the square root from the denominator.

Rationalising the Denominator

To rationalise the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator:

(6-4√3) / (6+4√3) × (6-4√3) / (6-4√3)

This simplifies to:

((6-4√3) × (6-4√3)) / ((6+4√3) × (6-4√3))

Expanding the Numerator and Denominator

Now, let's expand the numerator and denominator:

Numerator: (6-4√3) × (6-4√3) = 36 - 48√3 + 48√3 - 48 = 36 - 48 = -12

Denominator: (6+4√3) × (6-4√3) = 36 - 48√3 + 48√3 - 48 = 36 - 48 = -12

Simplifying the Expression

Now that we have simplified the numerator and denominator, we can rewrite the expression as:

(-12) / (-12)

Final Answer

The final answer is 1.

Conclusion

In this problem, we rationalised the denominator of the expression 6-4√3 divided by 6+4√3 by multiplying both the numerator and the denominator by the conjugate of the denominator. This resulted in a simplified expression of 1.

Tips and Tricks

When rationalising the denominator, always multiply both the numerator and the denominator by the conjugate of the denominator.
Make sure to simplify the numerator and denominator separately before rewriting the expression.
Rationalising the denominator is a crucial step in simplifying complex fractions, especially when dealing with square roots.

Common Mistakes

Failing to multiply both the numerator and the denominator by the conjugate of the denominator.
Not simplifying the numerator and denominator separately before rewriting the expression.
Not checking for any common factors in the numerator and denominator.

Real-World Applications

Rationalising the denominator has numerous real-world applications, including:

Simplifying complex fractions in algebra and calculus.
Solving equations and inequalities involving square roots.
Working with complex numbers and their applications in physics and engineering.

Practice Problems

Try these practice problems to test your skills in rationalising the denominator:

Rationalise the denominator of the expression 3+2√2 divided by 3-2√2.
Rationalise the denominator of the expression 4-3√5 divided by 4+3√5.

Solutions to Practice Problems

Rationalise the denominator of the expression 3+2√2 divided by 3-2√2: (3+2√2) / (3-2√2) × (3+2√2) / (3+2√2) = ((3+2√2) × (3+2√2)) / ((3-2√2) × (3+2√2)) = (9 + 12√2 + 12√2 + 8) / (9 - 12√2 + 12√2 - 8) = (17 + 24√2) / (1) = 17 + 24√2
Rationalise the denominator of the expression 4-3√5 divided by 4+3√5: (4-3√5) / (4+3√5) × (4-3√5) / (4-3√5) = ((4-3√5) × (4-3√5)) / ((4+3√5) × (4-3√5)) = (16 - 24√5 + 36√5 - 45) / (16 - 24√5 + 24√5 - 45) = (16 + 12√5 - 45) / (16 - 45) = (-29 + 12√5) / (-29) = 1 + (12√5) / (-29) = 1 - 12√5 / 29

Conclusion

Rationalising the denominator is a crucial step in simplifying complex fractions, especially when dealing with square roots. By multiplying both the numerator and the denominator by the conjugate of the denominator, we can eliminate the square root from the denominator and simplify the resulting expression. This has numerous real-world applications, including simplifying complex fractions in algebra and calculus, solving equations and inequalities involving square roots, and working with complex numbers and their applications in physics and engineering.

Introduction

In the previous article, we discussed how to rationalise the denominator of the expression 6-4√3 divided by 6+4√3. In this article, we will answer some frequently asked questions related to rationalising the denominator.

Q&A

Q1: What is rationalising the denominator?

A1: Rationalising the denominator is a process of eliminating the square root from the denominator of a fraction. This is done by multiplying both the numerator and the denominator by the conjugate of the denominator.

Q2: Why do we need to rationalise the denominator?

A2: We need to rationalise the denominator to simplify complex fractions, especially when dealing with square roots. This makes it easier to work with the expression and perform calculations.

Q3: How do I find the conjugate of the denominator?

A3: The conjugate of a binomial expression is obtained by changing the sign of the second term. For example, the conjugate of 6+4√3 is 6-4√3.

Q4: What is the difference between rationalising the denominator and simplifying the expression?

A4: Rationalising the denominator involves eliminating the square root from the denominator, while simplifying the expression involves reducing the fraction to its simplest form.

Q5: Can I rationalise the denominator of any expression?

A5: Yes, you can rationalise the denominator of any expression that has a square root in the denominator. However, you need to make sure that the numerator and denominator are both multiplied by the conjugate of the denominator.

Q6: How do I know if I have rationalised the denominator correctly?

A6: To check if you have rationalised the denominator correctly, multiply both the numerator and the denominator by the conjugate of the denominator and simplify the expression. If the resulting expression is in its simplest form, then you have rationalised the denominator correctly.

Q7: Can I use rationalising the denominator to solve equations and inequalities involving square roots?

A7: Yes, you can use rationalising the denominator to solve equations and inequalities involving square roots. This is because rationalising the denominator eliminates the square root from the denominator, making it easier to work with the expression.

Q8: Are there any real-world applications of rationalising the denominator?

A8: Yes, there are numerous real-world applications of rationalising the denominator, including simplifying complex fractions in algebra and calculus, solving equations and inequalities involving square roots, and working with complex numbers and their applications in physics and engineering.

Q9: Can I use rationalising the denominator to simplify complex fractions in algebra and calculus?

A9: Yes, you can use rationalising the denominator to simplify complex fractions in algebra and calculus. This is because rationalising the denominator eliminates the square root from the denominator, making it easier to work with the expression.

Q10: Are there any common mistakes to avoid when rationalising the denominator?

A10: Yes, there are several common mistakes to avoid when rationalising the denominator, including failing to multiply both the numerator and the denominator by the conjugate of the denominator, not simplifying the numerator and denominator separately, and not checking for any common factors in the numerator and denominator.

Conclusion

Rationalising the denominator is a crucial step in simplifying complex fractions, especially when dealing with square roots. By answering these frequently asked questions, we hope to have provided you with a better understanding of the process and its applications.

Tips and Tricks

Always multiply both the numerator and the denominator by the conjugate of the denominator.
Make sure to simplify the numerator and denominator separately before rewriting the expression.
Rationalising the denominator is a crucial step in simplifying complex fractions, especially when dealing with square roots.

Practice Problems

Try these practice problems to test your skills in rationalising the denominator:

Rationalise the denominator of the expression 3+2√2 divided by 3-2√2.
Rationalise the denominator of the expression 4-3√5 divided by 4+3√5.

Solutions to Practice Problems

Rationalise the denominator of the expression 3+2√2 divided by 3-2√2: (3+2√2) / (3-2√2) × (3+2√2) / (3+2√2) = ((3+2√2) × (3+2√2)) / ((3-2√2) × (3+2√2)) = (9 + 12√2 + 12√2 + 8) / (9 - 12√2 + 12√2 - 8) = (17 + 24√2) / (1) = 17 + 24√2
Rationalise the denominator of the expression 4-3√5 divided by 4+3√5: (4-3√5) / (4+3√5) × (4-3√5) / (4-3√5) = ((4-3√5) × (4-3√5)) / ((4+3√5) × (4-3√5)) = (16 - 24√5 + 36√5 - 45) / (16 - 24√5 + 24√5 - 45) = (16 + 12√5 - 45) / (16 - 45) = (-29 + 12√5) / (-29) = 1 + (12√5) / (-29) = 1 - 12√5 / 29