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Chapter 1: Problem 77

Will help you prepare for the material covered in the next section. Rationalize the denominator: \(\frac{7+4 \sqrt{2}}{2-5 \sqrt{2}}\)

Short Answer

Expert verified
-34/46 - 43*sqrt(2)/46

Step by step solution

01

Identify the Conjugate

The conjugate of the denominator \(2-5 \sqrt{2}\) is \(2 + 5 \sqrt{2}\). The conjugate is formed by changing the sign between the two terms.
02

Multiply by the Conjugate

Multiply both the numerator and denominator by the conjugate. \(\frac{7+4 \sqrt{2}}{2-5 \sqrt{2}} \times \frac{2 + 5 \sqrt{2}}{2 + 5 \sqrt{2}}\). This step allows us to remove the square root from the denominator.
03

Apply the Difference of Squares

When multiplying the denominators (which are conjugates), we can use the difference of squares formula, \(a^2 - b^2 = (a - b) \times (a + b)\). This gives us \((2^2 - (5 \sqrt{2})^2)\).
04

Multiply the Numerator

Distribute each term in the numerator to each term in the conjugate: \((7+4 \sqrt{2})(2 + 5\sqrt{2}) = 14 + 35\sqrt{2} + 8\sqrt{2} +20.\)
05

Simplify

Simplify the numerator and denominator. The numerator is \(14 + 35\sqrt{2} + 8\sqrt{2} +20 = 34 + 43\sqrt{2}\), the denominator is \(2^2 - (5 \sqrt{2})^2 = 4 - 50 = -46.\)
06

Multiply by \( -1 \)

To get the denominator as a positive number, you can multiply the numerator and denominator by \( -1 \), so \( -34 - 43\sqrt{2} = \frac{34}{46} + \frac{43\sqrt{2}}{46} \).

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