Surds

Surds are the numbers in the form of roots (√) to describe its exact value. The surds are irrational values because there are infinite number of non-recurring decimals.

You need to follow some rules to solve expressions that involve surds. One method is to rationalise the denominators and it’s done by ejecting the surd in denominator.

Sometimes it may be mandatory to find the greatest perfect square factor to solve surds. This is done by considering any possible factors of the value that is square rooted. For example, you need to solve for square root of 144. 2 x 72 gives 144 and we can have square root of 144 without a surd. Therefore we say that 144 is the greatest perfect square factor, since we cannot take the square root of a bigger number that can be multiplied by another to give 144.

Six Rules of Surds

Rule 1:

Surds

Example:

To simplify √18

18 = 9 x 2 = 32 x 2, since 9 is the greatest perfect square factor of 18.

Therefore, √18 = √32 x 2

= √32 x √2

= 3 √2

Rule 2:

Surds

Example:

√12 / 121 = √12 / √121

√22 x 3 / 11

Since 4 is the perfect square of 12

√22 x √3 / 11

= 2√3 / 11

Rule 3:

Surds

You can rationalize the denominator by multiplying the numerator and denominator by the denominator.

Example:

Rationalise

5/√7

5/√7 = 5/√7 x √7/√7

Multiply numerator and denominator by √7

= 5√7/7

Rule 4:

Surds

Example:

To simplify,

5√6 + 4√6

5√6 + 4√6 = (5 + 4) √6

by the rule

= 9√6

Rule 5:

Surds

Multiply top and bottom by a-b √n

This rule enables us to rationalise the denominator.

Example:

To Rationalise

Surds

Surds

Surds

Surds

Rule 6:

Surds

This rule enables you to rationalise the denominator.

Multiply top and bottom by a + b√n

Example:

To Rationalise

  

Surds

Surds

Surds

Surds

Now to start practicing problems and examples of surd based on above mentioned rules, please visit www.byjus.com


Practise This Question

Draw a line AB. At A, draw an arc of length 3cm using compass such that it intersects AB at O. With the same spread of compass, put the compass pointer at O and make an arc that intersects the previous arc at P. With the same spread again, put the compass pointer at P and draw an arc that intersects the first arc at Q. Join A and Q. Using the protractor, measure QAB. What is the value of QAB.