# Simplify the Surd (3+2sqrt(5))/sqrt(5): my answer is (3sqrt(5) + 10)/5

#### bumblebee123

##### New member
Hello,

I have a question on surds, please can you help. I need to simplify the following:

(3+2sqrt(5))/sqrt(5)

I am told to get rid of the surd in the denominator by multiplying numerator and denominator by the surd at the bottom as follows:

(3+2sqrt(5))/sqrt(5) x sqrt(5)/sqrt(5)

So I get

(3sqrt(5) + 2sqrt(5)sqrt(5))/5

then

(3sqrt(5) + 2x5)/5

(3sqrt(5) + 10)/5

The answer in the book is :

(2+3sqrt(5))/5

Why are our answers different. Can anyone tell me what I have done wrong please.

#### Denis

##### Senior Member
Book is wrong, you're correct.

You can "see" this by simply evaluating each version:
u = sqrt(5)

(3 + 2u) / u = 3.3416... (original)

(2 + 3u) / 5 = 1.7416... (book)

(3u + 10) / 5 = 3.3416... (yours)

#### ksdhart2

##### Full Member
Hmm... How curious. I often find that expressions written on this site are ambiguous or incorrect due to missing grouping symbols. However, in this particular case, the answer is incorrect because of too many grouping symbols. That is to say;

$$\displaystyle \left( 2 + 3\sqrt{5} \right) \div 5 = \dfrac{2 + 3\sqrt{5}}{5} \approx 1.7416$$

Exactly as Denis indicated. But:

$$\displaystyle 2 + 3\sqrt{5}\div 5 = 2 + \dfrac{3\sqrt{5}}{5} \approx 3.3416$$

So, if the grouping symbols in the book's answer are removed, it is also correct.

#### Dr.Peterson

##### Elite Member
$$\displaystyle \left( 2 + 3\sqrt{5} \right) \div 5 = \dfrac{2 + 3\sqrt{5}}{5} \approx 1.7416$$

Exactly as Denis indicated. But:

$$\displaystyle 2 + 3\sqrt{5}\div 5 = 2 + \dfrac{3\sqrt{5}}{5} \approx 3.3416$$

So, if the grouping symbols in the book's answer are removed, it is also correct.
If we now go back to the original question, supposing that the book's answer was just copied incorrectly, it is this:

I need to simplify the following: (3 + 2sqrt(5))/sqrt(5)

My answer is: (3sqrt(5) + 10)/5

The answer in the book is: 2 + 3sqrt(5)/5

Then my reply would be that we just have two different opinions about what constitutes a fully simplified form. The OP's answer is the simplest single fraction form; the book's answer has been further simplified (by dividing each term separately by 5) to make a "mixed fraction". Both are correct, unless the book has somewhere stated a rule for the goal in simplifying that requires the extra step.

#### bumblebee123

##### New member
Thank you, everyone, for your help.

The answer in the book definitely has 2+3srt(5) all being divided by 5.
I see from Denis answers that they vary so I will take the book as a typo. Thank you for clearing that up for me.