RATIONALISATION OF SURDS

Number system of Class 9

Any irrational number of the form RATIONALISATION OF SURDS is given a special name surd. Where ‘a’ is called radicand, it should always be a rational number. Also the symbol RATIONALISATION OF SURDS is called the radical sign and the index n is called order of the surd.

RATIONALISATION OF SURDS is read as ‘nth root a’.

Some Identical Surds :

(i)  For example, the rationalizing factor of √5 is √5 and rationalizing factor of ∛2 is ∛2^2 or ∛4. Since, √5 × √5 = 5 and ∛2 × ∛2^2 = ∛(2 × 2^2) = ∛2^3 = 2

(ii)  (a√z) × (b√z) = (a × b) × (√z × √z) = ab(√z)^2 = abz, which is rational. Therefore, each of the surds a√z and b√z is a rationalizing factor of the other.

(iii)  √5 × 2√5 = 2 × (√5)^2 = 3 × 5 = 15, which is rational. Therefore, each of the surds √5 and 2√5 is a rationalizing factor of the other.

(iv)  (√a + √b) × (√a - √b) = (√a)^2 - (√b)^2 = a - b, which is rational. Therefore, each of the surds (√a + √b)  and (√a - √b) is a rationalizing factor of the other.

(v)  (x√a + y√b) × (x√a - y√b) = (x√a)^2 - (y√b)^2 = ax - by, which is rational. Therefore, each of the surds (x√a + y√b) and (x√a - y√b) is a rationalizing factor of the other.

(vi)  (4√7 + √3) × (4√7 - √3) = (4√7)^2 - (√3)^2 = 112 - 3 = 109, which is rational. Therefore, each of the surd factors (4√7 + √3) and (4√7 - √3) is a rationalizing factor of the other.So the order of the surds and the radicands both should be same for similar surds.

(vii)  Also rationalizing factor of ∛(ab^2c^2) is ∛(a^2bc) because ∛(ab^2c^2) × ∛(a^2bc) = abc.

(b) Some Expression are not Surds:

Now we will see if the following surds are similar or dissimilar.

(i)RATIONALISATION OF SURDS

The first surd is RATIONALISATION OF SURDS which has the irrational factorRATIONALISATION OF SURDS   we have to check whether other surds have the same irrational factor or not.

(ii) The second surd is

RATIONALISATION OF SURDS

So the second surd can be reduced to RATIONALISATION OF SURDS which has the irrational factorRATIONALISATION OF SURDS

(iii) RATIONALISATION OF SURDS

The third surd doesn’t contain irrational factorRATIONALISATION OF SURDS and also the forth surds has the order 3, so the above set of four surds are dissimilar surds.

For checking the surds are similar or dissimilar, we need to reduce the surds irrational factor of the surds which is lowest among the surds and match with other surds if it is same, then we can call it as similar or dissimilar surds.

 

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