Surd Definition

The definition for surd says that if the number is a positive rational number where it should not be expressed as the nth power of the rational number, then the irrational number `root(n)(a)` or a `1/n` , which is the positive nth root of a, can be called as the surd. But the expressions such as `sqrt(2)` ,` root(3)(3) ` , etc.which cannot be written as the exact value of the numbers. These numbers are called the irrational numbers. Now we see the surd definition.

About surd and its definition:

Now we see about the types of surd's and its definition  and some of concepts with the help of the tutors.

Mixed surd's:

In a surd, if one factor number is rational number and the other factor is an irrational number then the surd can be called as the mixed surd.

Example: 2√5, `5/9` √15, -2 √3

Pure surds:

When the surd has unity and it consists of only rational factors then other factor being irrational number then it can be called pure surd.

Example: √8,  `1/3` ,5 `1/4`

Hence the surds are irrational numbers and they can be added or subtracted with the real numbers also. Also a rational numbers are added or subtracted from a surd. The resultant number will be a real number.

Problems for surd:

Example 1:

Find out the following numbers which are surd's or not?

1. `sqrt(7)`

2. `sqrt(12)`

3. `root(3)(27)`

4. `sqrt(5)`

5. `sqrt(225)`

6. `root(3)(17)`

Solution :

1. `sqrt(7)` is the surd because it get the decimal values.

2. `sqrt(12)` is a surd because it get decimal values

3. `root(3)(27)` is not a surd because it gives `root(3)(27) = 3`

4. `sqrt(5)` is a surd which gives the decimal value.

5. `sqrt(225)` is not a surd because it gives `sqrt(225) = 15`

6. `root(3)(17)` is a surd.


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Example 2:

Find the surd value for `sqrt(45)` ?

Solution:

Now we are going to find the surd value for the given number as follows,

The given number has one factor 9 only where it is a perfect square.So, it can be expressed as a product of rational number and as a surd.

`sqrt(45)` = `sqrt( 9 * 5)`

= `sqrt(9) * sqrt(5)`

= `3 * sqrt(5)`

= `3 sqrt(5)` This is called a mixed surd.