Surds are the numbers in the form of roots (âˆš) 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:

Example:

To simplify âˆš18

18 = 9 x 2 = 3^{2} x 2, since 9 is the greatest perfect square factor of 18.

Therefore, âˆš18 = âˆš3^{2} x 2

= âˆš3^{2 }x âˆš2

= 3 âˆš2

Rule 2:

Example:

âˆš12 / 121 = âˆš12 / âˆš121

âˆš2^{2} x 3 / 11

Since 4 is the perfect square of 12

âˆš2^{2} x âˆš3 / 11

= 2âˆš3 / 11

Rule 3:

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

Example:

Rationalise

5/âˆš7

5/âˆš7 = 5/âˆš7 x âˆš7/âˆš7

Multiply numerator and denominator by âˆš7

= 5âˆš7/7

Rule 4:

Example:

To simplify,

5âˆš6 + 4âˆš6

5âˆš6 + 4âˆš6 = (5 + 4) âˆš6

by the rule

= 9âˆš6

Rule 5:

Multiply top and bottom by a-b âˆšn

This rule enables us to rationalise the denominator.

Example:

To Rationalise

Rule 6:

This rule enables you to rationalise the denominator.

Multiply top and bottom by a + bâˆšn

Example:

To Rationalise

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