Can surds upon surds be simplified?

  • Thread starter Mentallic
  • Start date
  • Tags
    Simplifying
In summary, there are some surds inside of surds that can be converted into a simpler form, while others cannot. The quickest way to determine if an equation can be simplified is by checking the discriminant. However, there is a method called "denesting" that can be used to quickly check if an equation can be simplified, by checking if \sqrt{A^2-B^2} is a rational number. This method may not work for all equations, but it can provide a good indication.
  • #1
Mentallic
Homework Helper
3,802
95
For some surds inside of surds, they can be converted into a more simple form:

[tex]\sqrt{a+b\sqrt{c}}=e+f\sqrt{g}[/tex]

Such as: [tex]\sqrt{11-6\sqrt{2}}=3-\sqrt{2}[/tex]

However, there are some that cannot be simplified into this form (as far as I know).

Such as: [tex]\sqrt{3+\sqrt{7}}[/tex]

I am curious to know if there is fast method in realizing whether these types of equations can be simplified.
My only way of knowing so far is as follows:

To see if [tex]\sqrt{31+12\sqrt{3}}[/tex] can be simplified, first I let it be equal to some general simplified form:

[tex]\sqrt{31+12\sqrt{3}}=a+b\sqrt{3}[/tex]

squaring both sides:

[tex]a^2+3b^2+2\sqrt{3}ab=31+12\sqrt{3}[/tex]

Therefore, [tex]a^2+3b^2=31[/tex] (1) and
[tex]2\sqrt{3}ab=12\sqrt{3}[/tex] (2)

Making a or b the subject in (2)
[tex]b=\frac{6}{a}[/tex]

Substituting into (1)

[tex]a^2+3(\frac{36}{a^2})=31[/tex]

[tex]a^4-31a^2+108=0[/tex]

Now we have a quadratic in [tex]a^2[/tex]. I will now know from the quadratic formula if this expression can be simplified or not by looking at the discriminant. If it is a perfect square, then it can be simplified, else, it cannot be.
 
Mathematics news on Phys.org
  • #3
Thanks for the link :smile:

For simplicities sake, I have concluded from reading through the site that to quickly check if such questions as I have posed can be denested (this is the term used), I can shorten the procedure by checking:

Given [tex]\sqrt{A \pm B}[/tex] where [tex]A,B[/tex] all reals, [tex]A+B>0[/tex]

Checking to see if [tex]\sqrt{A^2-B^2}[/tex] is a rational number will give me the indication whether to pursue the simplified answer.

Of course there is a wider array of problems, but I am happy with the progress made for the moment.
 

1. What is a surd?

A surd is an irrational number that cannot be expressed as a ratio of two integers. It is usually represented by a root sign (√) followed by a non-perfect square number.

2. How do you simplify surds upon surds?

To simplify surds upon surds, we first break down the expression into individual surds and then use the properties of surds to simplify them. This involves finding the common factors and then rationalizing the denominators if necessary.

3. What are the properties of surds that can help simplify them?

Some of the properties of surds that are useful in simplifying them include the product property (√a * √b = √ab), the quotient property (√a / √b = √a/b), and the power property (√a ^n = √a * √a * ... * √a, n times).

4. Can surds upon surds be simplified to whole numbers?

It is possible for surds upon surds to simplify to whole numbers, but it depends on the specific expression. In some cases, the surds may cancel out completely, resulting in a whole number. However, in most cases, the expression will still contain surds after simplifying.

5. Are there any rules or guidelines to follow when simplifying surds upon surds?

Yes, there are certain rules and guidelines that should be followed when simplifying surds upon surds. These include rationalizing the denominator, simplifying the surds as much as possible, and checking for common factors. It is also important to pay attention to the signs and powers of the surds when simplifying.

Similar threads

Replies
5
Views
1K
  • General Math
Replies
15
Views
1K
Replies
7
Views
798
  • General Math
Replies
1
Views
1K
Replies
7
Views
3K
Replies
17
Views
3K
Replies
1
Views
695
Replies
4
Views
2K
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
11
Views
239
Back
Top