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Chapter 10: Problem 64

Rationalize each denominator. $$\frac{1+\sqrt{2}}{3+\sqrt{5}}$$

Short Answer

Expert verified
\[ \frac{3 - \, \sqrt{5} + 3 \, \sqrt{2} - \, \sqrt{10}}{4} \]

Step by step solution

01

Identify the Conjugate

The denominator of the given fraction is \[3 + \, \sqrt{5}\].To rationalize it, identify its conjugate, which changes the sign of the irrational term: \[3 - \, \sqrt{5}\].
02

Multiply Numerator and Denominator by the Conjugate

Multiply both the numerator and the denominator by the conjugate of the denominator: \[\frac{(1 + \, \sqrt{2})(3 - \, \sqrt{5})}{(3 + \, \sqrt{5})(3 - \, \sqrt{5})}\].This step ensures the denominator becomes a rational number.
03

Expand the Numerator

Distribute the terms in the numerator: \[(1 + \, \sqrt{2})(3 - \, \sqrt{5}) = 1 \, \cdot \, 3 + 1 \, \cdot \, (- \, \sqrt{5}) + \, \sqrt{2} \, \cdot \, 3 + \, \sqrt{2} \, \cdot \, (- \, \sqrt{5}) = 3 - \, \sqrt{5} + 3 \, \sqrt{2} - \, \sqrt{10}\]. Therefore, the expanded numerator is \[3 - \, \sqrt{5} + 3 \, \sqrt{2} - \, \sqrt{10}\].
04

Simplify the Denominator

Use the difference of squares formula to simplify the denominator: \[(3 + \, \sqrt{5})(3 - \, \sqrt{5}) = 3^2 - (\sqrt{5})^2 = 9 - 5 = 4\]. So, the simplified denominator is 4.
05

Combine the Results

Combine the simplified numerator and denominator to obtain the rationalized fraction: \[\frac{3 - \, \sqrt{5} + 3 \, \sqrt{2} - \, \sqrt{10}}{4}\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

conjugate
To rationalize a denominator containing a square root, we use the **conjugate** of the denominator. The conjugate of a binomial like \(3 + \sqrt{5}\) is formed by changing the sign of the square root term. So, the conjugate of \(3 + \sqrt{5}\) is \(3 - \sqrt{5}\). Multiplying by the conjugate helps in getting rid of the square root in the denominator, turning it into a rational number.
difference of squares
The **difference of squares** formula is key to rationalizing denominators. This formula is expressed as \((a + b)(a - b) = a^2 - b^2\). In our example, when we multiply \(3 + \sqrt{5}\) by its conjugate \(3 - \sqrt{5}\), we apply this formula: \((3)^2 - (\sqrt{5})^2 = 9 - 5 = 4\). This simplifies the denominator to a rational number.
simplifying radicals
The process of **simplifying radicals** involves removing the square roots from the denominator. By multiplying both the numerator and the denominator by the conjugate, we turn the radical expression into a simpler, rational form. For example, multiplying \(\frac{1 + \sqrt{2}}{3 + \sqrt{5}}\) by \(\frac{3 - \sqrt{5}}{3 - \sqrt{5}}\) transforms it because the conjugate cancels out the square roots in the denominator, simplifying it to 4.
expanding binomials
When **expanding binomials** like \((1 + \sqrt{2})(3 - \sqrt{5})\), use the distributive property (FOIL method). Multiply each term in the first binomial by each term in the second binomial: \(1 \cdot 3 + 1 \cdot (-\sqrt{5}) + \sqrt{2} \cdot 3 + \sqrt{2} \cdot (-\sqrt{5})\). This gives us \(3 - \sqrt{5} + 3\sqrt{2} - \sqrt{10}\). Simplifying results in a new expression for the numerator in the rationalized fraction.

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