(b) Rationalize The Denominator Of $\frac{1}{(\sqrt{2}+1)(\sqrt{3}-1)}$.
Introduction
Rationalizing the denominator of a complex fraction is a crucial step in simplifying and solving mathematical expressions. In this article, we will focus on rationalizing the denominator of the given fraction . We will use various techniques and strategies to simplify the expression and arrive at the final result.
Understanding the Problem
The given fraction is . To rationalize the denominator, we need to eliminate the square roots from the denominator. This can be achieved by multiplying the numerator and denominator by a suitable expression that will eliminate the square roots.
Step 1: Multiply the Numerator and Denominator by the Conjugate
To rationalize the denominator, we need to multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a binomial expression is . In this case, the conjugate of is .
\frac{1}{(\sqrt{2}+1)(\sqrt{3}-1)} \cdot \frac{(\sqrt{2}-1)(\sqrt{3}+1)}{(\sqrt{2}-1)(\sqrt{3}+1)}
Step 2: Simplify the Expression
Now, we can simplify the expression by multiplying the numerator and denominator.
\frac{(\sqrt{2}-1)(\sqrt{3}+1)}{(\sqrt{2}+1)(\sqrt{3}-1)(\sqrt{2}-1)(\sqrt{3}+1)}
Step 3: Cancel Out Common Factors
We can cancel out common factors in the numerator and denominator.
\frac{(\sqrt{2}-1)(\sqrt{3}+1)}{(\sqrt{2}+1)(\sqrt{3}-1)(\sqrt{2}-1)(\sqrt{3}+1)} = \frac{\sqrt{2}-1}{(\sqrt{2}+1)(\sqrt{3}-1)}
Step 4: Multiply the Numerator and Denominator by the Conjugate Again
To rationalize the denominator completely, we need to multiply the numerator and denominator by the conjugate of the denominator again.
\frac{\sqrt{2}-1}{(\sqrt{2}+1)(\sqrt{3}-1)} \cdot \frac{(\sqrt{2}+1)(\sqrt{3}+1)}{(\sqrt{2}+1)(\sqrt{3}+1)}
Step 5: Simplify the Expression Again
Now, we can simplify the expression again by multiplying the numerator and denominator.
\frac{(\sqrt{2}-1)(\sqrt{2}+1)(\sqrt{3}+1)}{(\sqrt{2}+1)(\sqrt{3}-1)(\sqrt{2}+1)(\sqrt{3}+1)}
Step 6: Cancel Out Common Factors Again
We can cancel out common factors in the numerator and denominator.
\frac{(\sqrt{2}-1)(\sqrt{2}+1)(\sqrt{3}+1)}{(\sqrt{2}+1)(\sqrt{3}-1)(\sqrt{2}+1)(\sqrt{3}+1)} = \frac{(\sqrt{2}-1)(\sqrt{2}+1)(\sqrt{3}+1)}{(\sqrt{2}+1)^2(\sqrt{3}^2-1^2)}
Step 7: Simplify the Expression Completely
Now, we can simplify the expression completely by evaluating the square roots and simplifying the expression.
\frac{(\sqrt{2}-1)(\sqrt{2}+1)(\sqrt{3}+1)}{(\sqrt{2}+1)^2(\sqrt{3}^2-1^2)} = \frac{(\sqrt{2}-1)(\sqrt{2}+1)(\sqrt{3}+1)}{(\sqrt{2}+1)^2(3-1)} = \frac{(\sqrt{2}-1)(\sqrt{2}+1)(\sqrt{3}+1)}{(\sqrt{2}+1)^2(2)} = \frac{(\sqrt{2}-1)(\sqrt{2}+1)(\sqrt{3}+1)}{2(\sqrt{2}+1)^2} = \frac{(\sqrt{2}-1)(\sqrt{2}+1)(\sqrt{3}+1)}{2(\sqrt{2}+1)^2} = \frac{(\sqrt{2}^2-1^2)(\sqrt{3}+1)}{2(\sqrt{2}+1)^2} = \frac{(2-1)(\sqrt{3}+1)}{2(\sqrt{2}+1)^2} = \frac{1(\sqrt{3}+1)}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)(\sqrt{2}+1)} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2} = \frac{\sqrt{3}+1}{2(\sqrt{2}+1)^2<br/>
**Rationalizing the Denominator of a Complex Fraction: Q&A**
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**Q: What is rationalizing the denominator?**
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A: Rationalizing the denominator is a process of eliminating the square roots from the denominator of a fraction. This is done by multiplying the numerator and denominator by a suitable expression that will eliminate the square roots.
**Q: Why is rationalizing the denominator important?**
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A: Rationalizing the denominator is important because it allows us to simplify complex fractions and make them easier to work with. It also helps to eliminate any square roots that may be present in the denominator, making it easier to perform calculations.
**Q: How do I rationalize the denominator of a complex fraction?**
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A: To rationalize the denominator of a complex fraction, you 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 $(\sqrt{2}+1)(\sqrt{3}-1)$, the conjugate is $(\sqrt{2}-1)(\sqrt{3}+1)$.
**Q: What is the conjugate of a binomial expression?**
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A: The conjugate of a binomial expression $a+b$ is $a-b$. For example, the conjugate of $(\sqrt{2}+1)$ is $(\sqrt{2}-1)$, and the conjugate of $(\sqrt{3}-1)$ is $(\sqrt{3}+1)$.
**Q: How do I multiply the numerator and denominator by the conjugate?**
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A: To multiply the numerator and denominator by the conjugate, you need to multiply the numerator and denominator by the conjugate of the denominator. For example, if the denominator is $(\sqrt{2}+1)(\sqrt{3}-1)$, you would multiply the numerator and denominator by $(\sqrt{2}-1)(\sqrt{3}+1)$.
**Q: What happens when I multiply the numerator and denominator by the conjugate?**
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A: When you multiply the numerator and denominator by the conjugate, the square roots in the denominator are eliminated, and the fraction becomes simpler.
**Q: Can I rationalize the denominator of a complex fraction with multiple square roots?**
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A: Yes, you can rationalize the denominator of a complex fraction with multiple square roots. You need to multiply the numerator and denominator by the conjugate of the denominator, which is the product of the conjugates of each square root.
**Q: How do I simplify the fraction after rationalizing the denominator?**
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A: After rationalizing the denominator, you can simplify the fraction by canceling out any common factors in the numerator and denominator.
**Q: What are some common mistakes to avoid when rationalizing the denominator?**
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A: Some common mistakes to avoid when rationalizing the denominator include:
* Not multiplying the numerator and denominator by the conjugate
* Not simplifying the fraction after rationalizing the denominator
* Not canceling out common factors in the numerator and denominator
**Q: Can I use a calculator to rationalize the denominator of a complex fraction?**
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A: Yes, you can use a calculator to rationalize the denominator of a complex fraction. However, it's always a good idea to check your work by hand to make sure you get the correct answer.
**Q: How do I know if I have rationalized the denominator correctly?**
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A: To know if you have rationalized the denominator correctly, you need to check that the square roots in the denominator have been eliminated and that the fraction has been simplified. You can also use a calculator to check your work.