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 \(\sqrt[2]{2}\), \(2\sqrt[2]{2}\), \(5\sqrt[2]{2}\), \(7\sqrt[2]{2}\) are similar surds as all the surds contain same irrational factor \(\sqrt[2]{2}\). So the order of the surds and the radicands both should be same for similar surds.
Consider the following surds \(2\sqrt[2]{3}\), \(4\sqrt[2]{27}\), \(7\sqrt[2]{243}\), \(5\sqrt[2]{75}\)
The above surds have different irrational factor or surd factor but they can be reduced to same irrational factor containing \(\sqrt[2]{3}\).
\(4\sqrt[2]{27}\) = \(4\sqrt[2]{9\times 3}\) = \(4\sqrt[2]{3^{2}\times 3}\)= \(12\sqrt[2]{3}\)
\(7\sqrt[2]{243}\) = \(7\sqrt[2]{81\times 3}\) = \(4\sqrt[2]{9^{2}\times 3}\) = \(36\sqrt[2]{3}\)
\(5\sqrt[2]{75}\) = \(5\sqrt[2]{25\times 3}\) = \(5\sqrt[2]{5^{2}\times 3}\) = \(25\sqrt[2]{3}\)
From the above example it can be seen that the first surd has the irrational factor \(\sqrt[2]{3}\), but other three surds which have irrational factors \(\sqrt[2]{27}\), \(\sqrt[2]{243}\), \(\sqrt[2]{75}\) respectively and can be reduced to \(\sqrt[2]{3}\). So the above surds are also similar surds.
More example,
(i) √5, 7√5, 10√5, -3√5, 5\(^{1/2}\), 10 ∙ √5, 12 ∙ 5\(^{1/2}\) are similar surds;
(ii) 7√5, 2√125, 5\(^{2/5}\)are similar surds since 2√125 = 2 ∙ \(\sqrt{5 ∙ 5 ∙ 5}\) = 2√5 and 5\(^{5/2}\) =\(\sqrt{5^{5}}\) = \(\sqrt{5 ∙ 5 ∙ 5 ∙ 5 ∙ 5}\) = 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 \(\sqrt[2]{3}\), \(2\sqrt[3]{3}\), \(5\sqrt[2]{6}\), \(7\sqrt[4]{3}\) are dissimilar surds as all the surds contain different irrational factors as \(\sqrt[2]{3}\), \(\sqrt[3]{3}\), \(\sqrt[2]{6}\), \(\sqrt[4]{3}\). 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.
\(3\sqrt[2]{3}\), \(4\sqrt[2]{12}\), \(5\sqrt[2]{18}\), \(7\sqrt[3]{3}\)
The first surd is \(3\sqrt[2]{3}\) which has the irrational factor \(\sqrt[2]{3}\), we have to check whether other surds have the same irrational factor or not.
The second surd is
\(4\sqrt[2]{12}\)= \(4\sqrt[2]{4\times 3}\)= \(4\sqrt[2]{2^{2}\times 3}\)= \(8\sqrt[2]{3}\)
So the second surd can be reduced to \(8\sqrt[2]{3}\) which has the irrational factor \(\sqrt[2]{3}\).
Now the third surd is
\(5\sqrt[2]{18}\)= \(5\sqrt[2]{9\times 2}\)= \(4\sqrt[2]{3^{2}\times 2}\)= \(12\sqrt[2]{2}\)
The third surd doesn’t contain irrational factor \(\sqrt[2]{3}\) 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, 7\(^{5/6}\) 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 = \(\sqrt[n]{4^{n}}\)
In general, if a he a rational number then,
x = √x\(^{2}\) = ∛x\(^{3}\) = ∜x\(^{4}\) = \(\sqrt[n]{x^{n}}\).
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