Problem 77 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Step 3: 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 \((2^2 - (5 \sqrt{2})^2)\).

Step 4: Multiply the Numerator

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

Step 5: Simplify

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

Step 6: Multiply by \( -1 \)

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