# Surd

Updated: 26 Aug 2023

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## 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

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

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

Order

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

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

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

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

7. \sqrt{3} is called _____ surd.
O Cubic
O None of these

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

8. \sqrt[3]{3} is called _____ surd.
O Cubic
O None of these

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

9. \sqrt[4]{3} is called _____ surd.
O Cubic
O None of these

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 None of these

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

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

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

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

Compound

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

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

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

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}