Quick way to simplify (12(sqrt(2) + 17)

[bigevil]

Homework Statement

This isn't really a problem at all. Have been working on some problems in Mary L Boas' textbook with some difficulty, but I have managed to solve the majority of the questions eventually. However, I've been stumped at how to arrive at the precise answers quoted in the book.

For instance, one of the answers is $$\frac{1}{32} (51\sqrt{2} - ln (1 + \sqrt{2})$$. But due to the method of arriving at the answer, my answer was $$\frac{1}{32}(51\sqrt{2}) - \frac{1}{128}(ln (17 + 2\sqrt{2})$$ which is equivalent (12 root 2 + 17 = (1 + rt 2)^4, after working backwards). But I'm wondering if there's actually a way to factorise or simplify my original (12 rt 2 + 17) quickly without working backwards.

 

[tiny-tim]

Homework Statement

This isn't really a problem at all. Have been working on some problems in Mary L Boas' textbook with some difficulty, but I have managed to solve the majority of the questions eventually. However, I've been stumped at how to arrive at the precise answers quoted in the book.

For instance, one of the answers is $$\frac{1}{32} (51\sqrt{2} - ln (1 + \sqrt{2})$$. But due to the method of arriving at the answer, my answer was $$\frac{1}{32}(51\sqrt{2}) - \frac{1}{128}(ln (17 + 2\sqrt{2})$$ which is equivalent (12 root 2 + 17 = (1 + rt 2)^4, after working backwards). But I'm wondering if there's actually a way to factorise or simplify my original (12 rt 2 + 17) quickly without working backwards.

Hi bigevil!

(have a square-root: √ )

This is a bit of a hindsight way too …

you could have noticed that (17 + 12√2)(17 - 12√2) = 289 - 288 = 1,

so any factor was going to have the same property …

like 1 ± √2 and 3 ± 4√2

 

[bigevil]

Thanks tim! That's pretty clever...