Questions & Answers

State which is surd is \[{\text{5 }}\sqrt[{\text{3}}]{{{\text{135}}}}\].
A. Pure surd
B. Mixed surd
C. Not a surd
D. None of the above

Answer Verified Verified
Hint: Let us split the number inside the cube root into its factor. So, we can check whether a given number is rational or irrational.

Complete step-by-step answer:
Now as we know that rational number is a number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number.
And, irrational numbers are the numbers that cannot be written as a simple fraction. Numbers which are not rational are irrational.
And surds are numbers left in 'square root form' (or 'cube root form' etc). They are therefore irrational numbers. The reason we leave them as surds is because in decimal form, they would go on forever and so this is a very clumsy way of writing them.
And, a surd having the rational coefficient other than unity is called a mixed surd. In other words, if some part of the quantity under the radical sign is taken out of it, then it makes a mixed surd.
So, some mixed surds are \[{\text{2 }}\sqrt[{\text{3}}]{3},{\text{ }}7{\text{ }}\sqrt[{\text{3}}]{{13}},{\text{ }}x\sqrt y \] and \[19\sqrt 2 \] etc.
And, a surd in which the whole of the rational number is under the radical sign and makes the radicand, is called pure surd. In other words, a surd having no rational factor except unity is called a pure surd or complete surd.
So, some mixed surds are \[\sqrt[{\text{3}}]{3},{\text{ }}\sqrt[{\text{3}}]{{13}},{\text{ }}\sqrt y \] and \[\sqrt {59} \] etc.
Now we have to find which type of surd is \[{\text{5 }}\sqrt[{\text{3}}]{{{\text{135}}}}\].
So, now splitting the number inside the cube root into factors so that cube root will be reduced.
So, \[{\text{5 }}\sqrt[{\text{3}}]{{{\text{135}}}}\] = \[{\text{5 }}\sqrt[{\text{3}}]{{{\text{5*3*3*3}}}}{\text{ = 5 }}\sqrt[{\text{3}}]{{{\text{5*}}{{\left( {\text{3}} \right)}^{\text{3}}}}}{\text{ = 15 }}\sqrt[{\text{3}}]{{\text{5}}}\].
So, now from the above definition of mixed surd. We can say that \[{\text{5 }}\sqrt[{\text{3}}]{{{\text{135}}}}\] is a mixed surd.
Hence, the correct option will be B.

Note: Whenever we come up with this type of problem then we should first write all the definitions included in the given options (here surd, mixed sure and pure surd). And then we had to split the term inside the root (cube or square) So, that it became easy to identify which type of surd is the given number.
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