Surd

Published: 26 Aug 2023

Among the vast fields of mathematics, surd hold a special place. Surds are irrational numbers in various mathematical expressions, equations, and real-world scenarios. This article is about the definition, properties, explanation, examples and MCQs about surds.

Definition of Surd

The irrational root of a rational number is called a surd. A number of the form of $ \sqrt[n]{a} $ is called Surd, where a is a positive rational number.

A number will be a surd, if

  • It is irrational
  • It is a root
  • A root of a rational number

Examples of Surds

$ \sqrt{3} $ and $ \sqrt{5+\sqrt{3}} $

  • In the above examples, both are irrational numbers.
  • The first number is a root of rational number $ 3 $, whereas the second number is a root of irrational number $ 5+\sqrt{3} $.
  • Thus, $ \sqrt{3} $ is a surd and $ \sqrt{5+\sqrt{3}} $ is not a surd.
  • $ \sqrt[3]{8} $ is not a surd because its value is $ 2$ which is rational.
  • $ \sqrt{-2} , \sqrt{-3} $ are not surds because $ -2 \ and \ -3 $ are negative.

Conjugate of Surds

  • The conjugate of $ a\sqrt{x}+b\sqrt{y} $ is $ a\sqrt{x}-b\sqrt{y} $.
  • Similarly the conjugate of $ 5+\sqrt{3} \ is \ 5-\sqrt{3} $

Surds MCQs

1. A number of the form of $ \sqrt[n]{a} $ is called ______, where a is a positive rational number.
O Conjugate
O Surd
O Rational
O None of these
Show Answer

Answer:
Surd


2. The rules for surd is/are _________
O It is irrational
O It is a root
O A root of a rational number.
O All of these
Show Answer

Answer:
All of these


3. In a surd, $ \sqrt[n]{a}, n $ is called ______ of the surd.
O type
O order
O conjugate
O None of these
Show Answer

Answer:
Order


4. $ \sqrt{3} $ is
O a surd
O not a surd
O conjugate of a surd
O None of these
Show Answer

Answer: a surd
Explanation: Here is the root of rational number, so it is a surd


5. $ \sqrt{5+\sqrt{3}} $ is
O a surd
O not a surd
O conjugate of a surd
O None of these
Show Answer

Answer: not a surd
Explanation: Here the root of irrational number, so it is not a surd.


6. $ \sqrt[3]{8} $ is
O a surd
O not a surd
O conjugate of a surd
O None of these
Show Answer

Answer: not a surd
Explanation:
$ \sqrt[3]{8} $ is not a surd because its value is 2 which is rational.


7. $ \sqrt{3} $ is called _____ surd.
O Quadratic
O Cubic
O Biquadratic
O None of these
Show Answer

Answer: Quadratic
Explanation: Here the surd index order is 2, so it is called quadratic.


8. $ \sqrt[3]{3} $ is called _____ surd.
O Quadratic
O Cubic
O Biquadratic
O None of these
Show Answer

Answer: Cubic
Explanation:
Here the surd index order is 3, so it is called cubic.


9. $ \sqrt[4]{3} $ is called _____ surd.
O Quadratic
O Cubic
O Biquadratic
O None of these
Show Answer

Answer: Biquadratic
Explanation:
Here the surd index order is 4, so it is called biquadratic.


10. $ \sqrt[3]{3} \ and \ \sqrt[3]{5} $ is called _____ surds.
O Quadratic
O equiradical
O Biquadratic
O None of these
Show Answer

Answer: equiradical
Explanation:
Surds of the same order are called equiradical surds which is 3.


11. $ \sqrt{3} $ is called _____ surd.
O Monomial
O Binomial
O Trinomial
O None of these
Show Answer

Answer: Monomial
Explanation:
Surd consists of one term is called monomial.


12. $ \sqrt{3}-2 $ is called _____ surd.
O Monomial
O Binomial
O Trinomial
O None of these
Show Answer

Answer: Binomial
Explanation:
Surd consists of Two terms is called binomial.


13. $ 2-\sqrt{3}+\sqrt{5} $ is called _____ surd.
O Monomial
O Binomial
O Trinomial
O None of these
Show Answer

Answer: Trinomial
Explanation:
Surd consists of three terms is called trinomial.


14. Expressions containing two or more surds are called _____ surds.
O Monomial
O Binomial
O Compound
O None of these
Show Answer

Answer:
Compound


15. $ \sqrt[3]{5} $ is called _____ surd.
O an entire
O mixed
O Trinomial
O None of these
Show Answer

Answer: an entire
Explanation:
If the coefficient of the surd is $ \pm1 $ is called an entire surd.


16. $ -4 \sqrt[3]{5} $ is called _____ surd.
O an entire
O mixed
O Trinomial
O None of these
Show Answer

Answer: mixed
Explanation:
If the coefficient of the surd is other than $ \pm1 $ is called mixed surd.


17. $ \sqrt{8} and \sqrt{18} $ are _____ surds.
O Similar
O Like
O Both a & b
O None of these
Show Answer

Answer: Both a & b
Explanation:
$ \sqrt{8}=\sqrt{4 \times 2} $
$ \sqrt{8}=2\sqrt{2} $
AND
$ \sqrt{18}=\sqrt{9 \times 2} $
$ \sqrt{18}=3\sqrt{2} $


18. Conjugate of $ 3-\sqrt{5} $ is
O $ -3-\sqrt{5} $
O $ -3+\sqrt{5} $
O $ 3+\sqrt{5} $
O $ \sqrt{3}+\sqrt{5} $
Show Answer

Answer: $ 3+\sqrt{5} $
Explanation:
The conjugate of
$ a\sqrt{x}+b\sqrt{y} $ is
$ a\sqrt{x}-b\sqrt{y} $

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jawad khalil
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