# How to simplify radicals in the denominator?

#### GrannySmith

##### New member
I have some problems I am stuck on. The goal here is to simplify radicals in the denominator. I understand that when there is a binomial in the denominator, you need to multiply both sides by the conjugate. For some reason though, I seem to be having trouble doing that or am making a mistake somewhere that I cannot figure out.

1. (8√6 + √10)/(2√2 - √7)

I multiply both sides by the conjugate and get (16√12 + 8√42 + 2√20 +√70)/1. Then when I simplify this I get 33√3 + 8√42 + 4√5 + √70. Seems like a long answer. What am I doing wrong?

#### MarkFL

Staff member
You are not expanding correctly. The second binomial has a negative sign that you aren't taking into account. Recall:

$$\displaystyle (a+b)(c-d)=ac-ad+bc-bd$$

#### GrannySmith

##### New member
I redid the problem like 2 times again but cannot catch on to my mistake. My mistake is in the denominator correct?

(2√2 - √7)(2√2 +√7)

I'm going to take this step by step.

(2√2)(2√2) = 4√4 = 8

(2√2)(√7) = 2√14

(-√7)(2√2) = -2√14

(-√7)(√7) = -√49 = -7

2√14 - 2√14 cancel each other out.

#### Opalg

##### MHB Oldtimer
Staff member
I have some problems I am stuck on. The goal here is to simplify radicals in the denominator. I understand that when there is a binomial in the denominator, you need to multiply both sides by the conjugate. For some reason though, I seem to be having trouble doing that or am making a mistake somewhere that I cannot figure out.

1. (8√6 + √10)/(2√2 - √7)

I multiply both sides by the conjugate and get (16√12 + 8√42 + 2√20 +√70)/1. Then when I simplify this I get 33√3 + 8√42 + 4√5 + √70. Seems like a long answer. What am I doing wrong?
That looks correct to me.

#### GrannySmith

##### New member
That looks correct to me.
I redid this problem and cannot find anything wrong.

It's just that i have this gut feeling that this answer is wrong for some reason. Way longer than any of my past answers for other problems like this. If it's right though then I guess there's nothing I can do to simplify that!

#### Prove It

##### Well-known member
MHB Math Helper
I have some problems I am stuck on. The goal here is to simplify radicals in the denominator. I understand that when there is a binomial in the denominator, you need to multiply both sides by the conjugate. For some reason though, I seem to be having trouble doing that or am making a mistake somewhere that I cannot figure out.

1. (8√6 + √10)/(2√2 - √7)

I multiply both sides by the conjugate and get (16√12 + 8√42 + 2√20 +√70)/1. Then when I simplify this I get 33√3 + 8√42 + 4√5 + √70. Seems like a long answer. What am I doing wrong?
Why do you think having a long answer makes it incorrect?

#### MarkFL

Staff member
I have some problems I am stuck on. The goal here is to simplify radicals in the denominator. I understand that when there is a binomial in the denominator, you need to multiply both sides by the conjugate. For some reason though, I seem to be having trouble doing that or am making a mistake somewhere that I cannot figure out.

1. (8√6 + √10)/(2√2 - √7)

I multiply both sides by the conjugate and get (16√12 + 8√42 + 2√20 +√70)/1. Then when I simplify this I get 33√3 + 8√42 + 4√5 + √70. Seems like a long answer. What am I doing wrong?
You are not expanding correctly. The second binomial has a negative sign that you aren't taking into account. Recall:

$$\displaystyle (a+b)(c-d)=ac-ad+bc-bd$$
My apologies...I misread the expression you gave as the product of numerator and the conjugate, not the original. #### Opalg

##### MHB Oldtimer
Staff member
I redid this problem and cannot find anything wrong.

It's just that i have this gut feeling that this answer is wrong for some reason. Way longer than any of my past answers for other problems like this. If it's right though then I guess there's nothing I can do to simplify that!
The radicals in the original problem all contain different prime factors (3, 5, 2, 7). So you cannot expect them to combine in any way that will give you fewer than four terms in the answer.