Simplifying Surds: Adding and Subtracting

[V-Tec]

Ok here is my problem, I am not sure how to simplify surds with additions and subtractions in them such as ones like this:

http://img212.imageshack.us/img212/474/math010iz.gif

At the moment I have managed to simplify it to this:

http://img212.imageshack.us/img212/2011/math026eg.gif

However it is not simplified as possible, can someone give me some words of wisdom? Any help greatly appreciated!

 

[EnumaElish]

In this case, 80 = 4 x 20, so $\sqrt{80}=\sqrt 4 \times\sqrt {20}=2\sqrt {20}$.

Therefore $\sqrt{20}+\sqrt{80}= (1+2)\sqrt{20}=3\sqrt{20}$, which is $3\times 2\times\sqrt 5 = 6\sqrt 5$.

 

[quicknote]

I can provide you with a step-by-step approach to simplify surds with additions and subtractions. First, let's define what a surd is. A surd is a number that cannot be simplified to a rational number (a number that can be expressed as a ratio of two integers). It is usually represented by a root symbol, such as √2 or √3.

To simplify surds with additions and subtractions, we can use the following rules:

1. Simplify the individual surds within the expression first. For example, in the expression √12 + √27, we can simplify √12 to 2√3 and √27 to 3√3.

2. Combine like terms. In the above example, we can then combine 2√3 and 3√3 to get 5√3.

3. If there are no like terms, we can use the distributive property to simplify the expression. For example, in the expression 2√3 + √6, we can rewrite it as 2(√3) + √(2*3), which can then be simplified to 2√3 + √6.

4. If there are surds in the denominator, we can rationalize the denominator by multiplying both the numerator and denominator by the surd in the denominator. For example, in the expression √3/√5, we can multiply both the numerator and denominator by √5 to get (√3*√5)/(√5*√5), which simplifies to √15/5.

Using these rules, we can simplify expressions with surds and additions/subtractions. In your example, we can first simplify the individual surds:

√(8+2√7) = √8 + √(2√7)

We can then use the distributive property to write √(2√7) as √(2*7) or √14:

√8 + √(2√7) = √8 + √(2*7) = √8 + √14

Next, we can combine like terms by rewriting √8 as 2√2 (since √8 = √(4*2) = 2√2):