Surds

Surds

Surd is a number that does not have the exact value of square root of a rational number. All irrational numbers are surds. For example: √3, √5, √(2/7) etc. are surds.

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Types of surds

Pure surds: A surd which has only one irrational number such as √3, √13, etc is called pure surd.

Mixed surd (compound surd):The surd which has the product of rational and irrational number such as 5√2, 2√3, etc is called a mixed surd.

Like surd: Two or more surds of the same order are called like surds such as 2√3, 3√5.

Unlike surd: Two or more surds, which have different order are called unlike surds. For example,

Rationalization of surd

Surd can be rationalized by a certain surd to remove the root sign in surd. Look at the following examples:

a.  √2 is a surd. When we multiply by √2, then the product is in rational form. i.e. √2×√2 = 2 which is a rational number.

b.  When (√3 - √2) is multiplied by √3 + √2 then the product is in ratinal form. i.e. (√3 - √2) (√3 + √2) = 3 – 2 = 1 which is rational number. So, (√3+√2) is rationalization factor or conjugate factor of (√3-√2).

Rules of surds

1. Addition and subtraction:

Like surds can be added or subtracted. For example:

i.    5√3 + 2√3 = 7√3

ii.   7√2 - 3√2 = 4√2

2. Multiplication and division:

When two or more surds have the same order, we multiply or divide the internal value of surds. For example:

i.    √5 × √2 = √(5×2) = √10

ii.   √10 ÷ √2 = √(10÷2) = √5

Properties of surds

i.         The sum or difference of a rational and an irrational number is a irrational number. For example: (3 + √5) and (4 - √2)  are irrational numbers.

ii.        The product of a rational and irrational number is always an irrational number. It is also called a mixed surd. For example: 3×√2 = 3√2 is an irrational number.

iii.       If a+√b = c+√d, where a and c are rational numbers and √b and √d are irrational numbers, then a = c and b = d.

iv.       If a+m√b = c+n√b, where a, c, m, n are rational numbers and √b is an irraitonal number, then a = c and m = n.

Workout Examples

Example 1: Identify the following pure or mixed surds.

Example 2: Write the rationalizing factor of:

a.    √8

b.    √3+√2

c.     5√3 - 1

Solution:  a. √8

            = √2×2×2

            = 2√2 

            ∴ Rationalizing factor = √2

        b. √3 +√2

            ∴ Rationalizing factor = √3 - √2

        c. 5√3 - 1

           ∴ Rationalizing factor = 5√3 + 1

Example 3: Rationalize the denominator of the following:

Example 4: Simplify the following

You can comment your questions or problems regarding the surds here.