Question

What are like surds and unlike surds

Answer 1

Dear Student,
SOLUTION:

●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 2–√222, 22–√2222, 52–√2522, 72–√2722 are similar surds as all the surds contain same irrational factor 2–√222. So the order of the surds and the radicands both should be same for similar surds. 

Consider the following surds 23–√2232, 427−−√24272, 7243−−−√272432, 575−−√25752

The above surds have different irrational factor or surd factor but they can be reduced to same irrational factor containing 3–√232. 

427−−√24272 = 49×3−−−−√249×32 = 432×3−−−−−√2432×32= 123–√21232

7243−−−√272432 = 781×3−−−−−√2781×32 = 492×3−−−−−√2492×32 = 363–√23632

575−−√25752 = 525×3−−−−−√2525×32 = 552×3−−−−−√2552×32 = 253–√22532

From the above example it can be seen that the first surd has the irrational factor 3–√232, but other three surds which have irrational factors 27−−√2272, 243−−−√22432, 75−−√2752 respectively and can be reduced to 3–√232. So the above surds are also similar surds.

More example,

(i) √5, 7√5, 10√5, -3√5, 51/21/2, 10 ∙ √5, 12 ∙ 51/21/2are similar surds;

(ii) 7√5, 2√125, 52/52/5are similar surds since 2√125 = 2 ∙ 5∙5∙5−−−−−−√5∙5∙5 = 2√5 and 55/25/2 =55−−√55 = 5∙5∙5∙5∙5−−−−−−−−−−−−√5∙5∙5∙5∙5 = 25√5 i.e., each of the given surds can be expressed with the same surd-factor √5.

 

●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 don’t have same surd factor or can’t be reduced to same surd factor, then surds are called as dissimilar surds. For example 3–√232, 23–√3233, 56–√2562, 73–√4734 are dissimilar surds as all the surds contain different irrational factors as 3–√232, 3–√333, 6–√262, 3–√434. If the order of the surds or the radicands are different or can’t 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. 

33–√2332, 412−−√24122, 518−−√25182, 73–√3733

The first surd is 33–√2332 which has the irrational factor 3–√232, we have to check whether other surds have the same irrational factor or not.

The second surd is 

412−−√24122= 44×3−−−−√244×32= 422×3−−−−−√2422×32= 83–√2832

So the second surd can be reduced to 83–√2832which has the irrational factor 3–√232.

Now the third surd is

518−−√25182= 59×2−−−−√259×22= 432×2−−−−−√2432×22= 122–√21222

The third surd doesn’t contain irrational factor 3–√232 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, √2, 9√3, 8√5, ∛6, ∜17, 75/65/6 are unlike surds.

【Note】: A given rational number can be expressed in the form of a surd of any desired order.

For example, 4 = √16 = ∛64 = ∜256 = 4n−−√n4nn

In general, if a he a rational number then,

x = √x22 = ∛x33 = ∜x44 = xn−−√nxnn.




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Answer 2

Step-by-step explanation:

surds having the same irrational factors