Find the square root of $14+6\sqrt5 $

Question

Find the square root of $14+6\sqrt5$

I don't just want the answer, can you please tell me the method how you get the answer

You can check Denesting radicals at Wikipedia, You can find here several posts on similar problems on this site: Strategies to denest nested radicals (and the posts linked there), Denesting a square root: $\sqrt{7 + \sqrt{14}}$ (and the posts linked there). — Martin Sleziak, Sep 02 '17 at 08:37
You can probably find a few more posts about similar problems in the nested-radicals tag. And you can use Approach0 to search for some other similar problems. — Martin Sleziak, Sep 02 '17 at 08:38

score 2 · Answer 1 · answered Sep 02 '17 at 07:00

2

Hint:

$$(a+b \sqrt{5})^2 = 14 + 6 \sqrt{5}$$

Solve for $a$ and $b$.

answered Sep 02 '17 at 07:00

Siong Thye Goh

149,520
20
88
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Kindly fetch me complete solution, I'm bit confused – user476799 Sep 02 '17 at 07:02
On this site, you have to show us your effort first. Try to expand the quadratic term first and compare the coefficients. I can further tell you that $a$ and $b$ are integers. – Siong Thye Goh Sep 02 '17 at 07:04
My expansion says $a$ =349. And $b$=108. I'm confused – user476799 Sep 02 '17 at 07:06
How did you obtain $a=349$? what do you get upon expanding $(a+b \sqrt{5})^2$? – Siong Thye Goh Sep 02 '17 at 07:07

score 2 · Answer 2 · answered Sep 02 '17 at 07:06

As you expect the solution to be nice (as in you don't just write $\sqrt{14+6\sqrt{5}}$), one can use "creative writing" to receive a square via the binomial formula:

$$14+6\sqrt{5}=9+5+6\sqrt{5}=9+6\sqrt{5}+5=9+2\cdot 3\cdot\sqrt{5}+\sqrt{5}^2$$

I'm sure you can take it from here!

score 2 · Answer 3 · answered Sep 02 '17 at 07:07

2

Hint:

$$6\sqrt5=2\cdot3\cdot\sqrt5$$

Compare with $$2ab$$ in $$a^2+b^2+2ab$$

answered Sep 02 '17 at 07:07

lab bhattacharjee

274,582

Find the square root of $14+6\sqrt5 $

3 Answers3