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SURDS

Rational Numbers

A rational number is a number which can be written in the form p/q where p and q are integers and q should be greater than zero. P and q must not have a common factor between then except 1.

Examples of rational numbers is like 1/5, 3/7, 4/11, etc.

Irrational numbers

Irrational are numbers are numbers that cannot be written in the form p/q. Irrational numbers cannot be expressed as simple fraction.

Examples of irrational numbers includes:

π  √2   √3   √7  etc.

when irrational numbers are expressed as decimals, the decimals continues without end and without recurring

for example √2 = 1.414213562……… and π =3.141592654…..

surds

The roots of rational numbers that gives irrational numbers are called surds.

numbers under square-root sign that will result to a whole number after square-root operations are NOT surds.

a surd is an irrational number of the form ±√x such that ±√x cannot be written as a/b where a, b ∈ ℤ and b ≠ 0.

for example ∛64 is not surd because it will evaluate to 4 which is a whole number.

Simplifying surds

In order to simplify surds, the number under the root sign should be expressed as a product of two factors such that one factor is a perfect root.

Examples:

simplify the following

(a)  √12  (b) √32    (c)  √(3/4)   (d) ∛250

solution

(a) We first express 12 as a product of two values, where one value is a perfect square as shown

Then we separate the two values under the root sign to have two roots multiplied such as:

we know square root of 4 is 2, but squareroot of 3 is irrational, hence we write:

and finally we remove the multiplication sign X to have

(b) 32 can be expressed as a product of many factors:

  • 2x 16
  • 4 x 8
  • 2 x 2 x 2 x 2 x 2 = 25
  • we write 24 x 21

24 is a perfect square and so we can it can be simplified when inside the square-root sign.

however, the shortest way is to get the root of 16 and root of 2 because root of 16 exists and square-root of 2 is irrational. hence we write

square-root of 16=4 and squareroot of 2 is irrational, hence we have:

(c) here were are looking for square-root of resultant quotient from dividing 3 and 4.

but we can as well get the same result by dividing root of 3 with root of 4 as shown:

and this can be simplified to:

(d) This problem requires us to find the cube-root of 250. There is no whole number that can be a cube-root of 250.

factors of 250 includes:

  • 10 x 25
  • 5 x 50
  • 125 x 2

but 125 = 5 x 5 x 5 = 53

hence we can express cube-root of 250 as the cube-root of the product 2 x 125.

which is broken down to:

and from the laws of indices:

and hence the final expression becomes:


surds can also be expressed as single compound surd from other surds

for example:

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