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We will discuss about similar and dissimilar surds and their definitions.
Definition of Similar Surds:
Two or more surds are said to be similar or like surds if they have the same surd-factor.
or,
Two or more surds are said to be similar or like surds if they can be so reduced as to have the same surd-factor.
For example 2β2, 22β2, 52β2, 72β2 are similar surds as all the surds contain same irrational factor 2β2. So the order of the surds and the radicands both should be same for similar surds.
Consider the following surds 22β3, 42β27, 72β243, 52β75
The above surds have different irrational factor or surd factor but they can be reduced to same irrational factor containing 2β3.
42β27 = 42β9Γ3 = 42β32Γ3= 122β3
72β243 = 72β81Γ3 = 42β92Γ3 = 362β3
52β75 = 52β25Γ3 = 52β52Γ3 = 252β3
From the above example it can be seen that the first surd has the irrational factor 2β3, but other three surds which have irrational factors 2β27, 2β243, 2β75 respectively and can be reduced to 2β3. So the above surds are also similar surds.
More example,
(i) β5, 7β5, 10β5, -3β5, 51/2, 10 β β5, 12 β 51/2 are similar surds;
(ii) 7β5, 2β125, 52/5are similar surds since 2β125 = 2 β β5β5β5 = 2β5 and 55/2 =β55 = β5β5β5β5β5 = 25β5 i.e., each of the given surds can be expressed with the same surd-factor β5.
Definition of Dissimilar Surds:
Two or more surds are said to be dissimilar or unlike when they are not similar.
If two or more surds donβt have same surd factor or canβt be reduced to same surd factor, then surds are called as dissimilar surds. For example 2β3, 23β3, 52β6, 74β3 are dissimilar surds as all the surds contain different irrational factors as 2β3, 3β3, 2β6, 4β3. If the order of the surds or the radicands are different or canβt be reduced to a surd with same order and radicand, the surds will be dissimilar surds.
Now we will see if the following surds are similar or dissimilar.
32β3, 42β12, 52β18, 73β3
The first surd is 32β3 which has the irrational factor 2β3, we have to check whether other surds have the same irrational factor or not.
The second surd is
42β12= 42β4Γ3= 42β22Γ3= 82β3
So the second surd can be reduced to 82β3 which has the irrational factor 2β3.
Now the third surd is
52β18= 52β9Γ2= 42β32Γ2= 122β2
The third surd doesnβt contain irrational factor 2β3 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.
More example, β2, 9β3, 8β5, β6, β17, 75/6 are unlike surds.
Note: A given rational number can be expressed in the form of a surd of any desired order.
For example, 4 = β16 = β64 = β256 = nβ4n
In general, if a he a rational number then,
x = βx2 = βx3 = βx4 = nβxn.
11 and 12 Grade Math
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