1/(2+√3)+2/(5-√3) Find The Value
Introduction
In this article, we will delve into the world of mathematical expressions and explore the process of simplifying complex fractions involving square roots. The given expression, 1/(2+√3)+2/(5-√3), may seem daunting at first, but with a step-by-step approach, we can break it down and find its value.
Understanding the Expression
The given expression consists of two fractions: 1/(2+√3) and 2/(5-√3). To simplify this expression, we need to rationalize the denominators of both fractions. Rationalizing the denominator involves multiplying the numerator and denominator by a specific value to eliminate the square root from the denominator.
Rationalizing the Denominators
To rationalize the denominator of the first fraction, 1/(2+√3), we can multiply both the numerator and denominator by the conjugate of the denominator, which is (2-√3). This will eliminate the square root from the denominator.
\frac{1}{2+\sqrt{3}} \cdot \frac{2-\sqrt{3}}{2-\sqrt{3}} = \frac{2-\sqrt{3}}{(2+\sqrt{3})(2-\sqrt{3})}
Using the difference of squares formula, (a+b)(a-b) = a^2 - b^2, we can simplify the denominator:
(2+\sqrt{3})(2-\sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1
So, the first fraction becomes:
\frac{2-\sqrt{3}}{1} = 2-\sqrt{3}
Similarly, to rationalize the denominator of the second fraction, 2/(5-√3), we can multiply both the numerator and denominator by the conjugate of the denominator, which is (5+√3).
\frac{2}{5-\sqrt{3}} \cdot \frac{5+\sqrt{3}}{5+\sqrt{3}} = \frac{2(5+\sqrt{3})}{(5-\sqrt{3})(5+\sqrt{3})}
Using the difference of squares formula, (a+b)(a-b) = a^2 - b^2, we can simplify the denominator:
(5-\sqrt{3})(5+\sqrt{3}) = 5^2 - (\sqrt{3})^2 = 25 - 3 = 22
So, the second fraction becomes:
\frac{2(5+\sqrt{3})}{22} = \frac{10+2\sqrt{3}}{22}
Simplifying the Expression
Now that we have rationalized the denominators of both fractions, we can simplify the expression by combining the two fractions.
1/(2+√3)+2/(5-√3) = (2-\sqrt{3}) + \frac{10+2\sqrt{3}}{22}
To add these two fractions, we need to have a common denominator, which is 22.
(2-\sqrt{3}) = \frac{44-22\sqrt{3}}{22}
Now, we can add the two fractions:
\frac{44-22\sqrt{3}}{22} + \frac{10+2\sqrt{3}}{22} = \frac{44-22\sqrt{3}+10+2\sqrt{3}}{22}
Simplifying the numerator, we get:
44-22\sqrt{3}+10+2\sqrt{3} = 54-20\sqrt{3}
So, the simplified expression is:
\frac{54-20\sqrt{3}}{22}
Conclusion
In this article, we simplified the complex expression 1/(2+√3)+2/(5-√3) by rationalizing the denominators of both fractions and combining them into a single fraction. The final simplified expression is 54-20√3/22.
Final Answer
The final answer is .
Introduction
In our previous article, we explored the process of simplifying complex fractions involving square roots. The given expression, 1/(2+√3)+2/(5-√3), may seem daunting at first, but with a step-by-step approach, we can break it down and find its value. In this Q&A article, we will address some common questions and provide additional insights into the simplification process.
Q: What is the purpose of rationalizing the denominators?
A: Rationalizing the denominators is a crucial step in simplifying complex fractions involving square roots. By multiplying the numerator and denominator by the conjugate of the denominator, we can eliminate the square root from the denominator, making it easier to simplify the expression.
Q: Why do we need to multiply by the conjugate of the denominator?
A: The conjugate of the denominator is used to eliminate the square root from the denominator. When we multiply the numerator and denominator by the conjugate, we are essentially multiplying by 1, but in a form that allows us to simplify the expression.
Q: Can we simplify the expression further?
A: Yes, we can simplify the expression further by combining the two fractions. To add these two fractions, we need to have a common denominator, which is 22.
Q: How do we add the two fractions?
A: To add the two fractions, we need to add the numerators while keeping the common denominator. This involves adding the coefficients of the terms with the same variable.
Q: What is the final simplified expression?
A: The final simplified expression is 54-20√3/22.
Q: Can we simplify the expression further?
A: Yes, we can simplify the expression further by dividing the numerator and denominator by their greatest common divisor, which is 2.
Q: What is the final simplified expression after dividing by 2?
A: The final simplified expression after dividing by 2 is 27-10√3/11.
Q&A Session
Q: I'm having trouble understanding the concept of rationalizing the denominators. Can you provide an example?
A: Let's consider the fraction 1/(2+√3). To rationalize the denominator, we can multiply both the numerator and denominator by the conjugate of the denominator, which is (2-√3).
Q: I'm not sure how to multiply the numerator and denominator by the conjugate. Can you provide a step-by-step guide?
A: To multiply the numerator and denominator by the conjugate, we can follow these steps:
- Multiply the numerator and denominator by the conjugate: (2-√3)
- Simplify the numerator: 1(2-√3) = 2-√3
- Simplify the denominator: (2+√3)(2-√3) = 4 - 3 = 1
Q: I'm having trouble understanding the concept of combining fractions. Can you provide an example?
A: Let's consider the fractions 1/(2+√3) and 2/(5-√3). To combine these fractions, we need to have a common denominator, which is 22.
Q: I'm not sure how to combine the fractions. Can you provide a step-by-step guide?
A: To combine the fractions, we can follow these steps:
- Multiply the first fraction by 22/22 to get a common denominator: (2-√3)(22/22) = (44-22√3)/22
- Multiply the second fraction by 22/22 to get a common denominator: (10+2√3)(22/22) = (220+44√3)/22
- Add the two fractions: (44-22√3)/22 + (220+44√3)/22 = (264-22√3)/22
Conclusion
In this Q&A article, we addressed some common questions and provided additional insights into the simplification process of the complex expression 1/(2+√3)+2/(5-√3). We hope that this article has helped to clarify any confusion and provided a better understanding of the concept of rationalizing the denominators and combining fractions.