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Chapter 11: Problem 49

Simplify as completely as possible. (Assume \(x \geq 0 .)\) $$\frac{3}{3-\sqrt{2}}$$

Short Answer

Expert verified
\frac{9 + 3\sqrt{2}}{7}

Step by step solution

01

Identify the Conjugate

To simplify the expression, identify the conjugate of the denominator. The conjugate of \(3 - \sqrt{2}\) is \(3 + \sqrt{2}\).
02

Multiply by the Conjugate

Multiply both the numerator and the denominator by the conjugate \(3 + \sqrt{2}\). This gives us: \[\frac{3}{3 - \sqrt{2}} \times \frac{3 + \sqrt{2}}{3 + \sqrt{2}} = \frac{3(3 + \sqrt{2})}{(3 - \sqrt{2})(3 + \sqrt{2})}\]
03

Simplify the Numerator

Expand the numerator: \(3(3 + \sqrt{2}) = 9 + 3\sqrt{2}\).
04

Simplify the Denominator

Use the difference of squares formula \(a^2 - b^2 = (a + b)(a - b)\) to simplify the denominator: \[(3 - \sqrt{2})(3 + \sqrt{2}) = 3^2 - (\sqrt{2})^2 = 9 - 2 = 7\]
05

Combine the Results

Now, write the simplified form of the expression: \[\frac{9 + 3\sqrt{2}}{7}\]

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Key Concepts

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

Rationalizing the Denominator
Rationalizing the denominator is a process used to eliminate a radical, like a square root, from the bottom part of a fraction. This makes the fraction easier to work with and understand. To do this, we multiply both the numerator and the denominator by a value that will cancel out the radical. In the exercise, we multiplied by the conjugate of the denominator. This helps because the product of a number and its conjugate gets rid of the square root. For example, given \(\frac{3}{3 - \sqrt{2}}\), we multiply by \( \frac{3 + \sqrt{2}}{3 + \sqrt{2}}\) to simplify the radical in the denominator.
Difference of Squares Formula
The difference of squares formula is a handy algebraic tool. It helps to simplify expressions where you multiply two binomials that are the same except for a different sign between their middle terms. The formula is \(a^2 - b^2 = (a + b)(a - b)\). In the exercise, the terms \(3 - \sqrt{2}\) and \(3 + \sqrt{2}\) can be seen as \(a - b\) and \(a + b\). This allows us to use the formula to turn the product into a much simpler form. Specifically, \( (3 - \sqrt{2})(3 + \sqrt{2}) = 3^2 - (\sqrt{2})^2\). Solving that, we get 9 - 2, which equals 7. This makes our calculation less complex.
Conjugates in Algebra
Conjugates in algebra refer to pairs of binomials that have identical terms but opposite middle signs. For example, the conjugate of \(3 - \sqrt{2}\) is \(3 + \sqrt{2}\). Conjugates are very useful when working with radicals. Multiplying a number by its conjugate allows you to remove the radical part. This technique simplifies the expressions and makes them easier to use in further calculations. In our exercise, multiplying by the conjugate made the denominator a simple whole number, which streamlined the fraction: \( \frac{3}{3 - \sqrt{2}}\) became \( \frac{9 + 3\sqrt{2}}{7}\).

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