Mixed Surd Examples

Sir I am teaching surds to the students but I have some problems in the mixed surd so can you tell me that what is the mixed surd and the example

[Pathak]

Hey buddy a surd is an irrational number, a number which cannot be expressed as a fraction or as a terminating or recurring decimal. It is left as a square root.

It can also be a non-cube number left in cube root form and so on. Number. Decimal.

Mixed Surd:

Surds such as 2√3, 3√9 are called mixed surds as they containrational numbers such as 2, 3 and surds such as √3 and √9
I. A mixed surd can be expressed in the form of a pure surd.

For example,

(i) 3√5 = 32⋅5−−−−√
= 9⋅5−−−√

= √45

(ii) 4 ∙ ∛3 = 43−−√3
∙ ∛3 = 64−−√3 ∙ ∛3 = 64−−√3⋅3

= ∛192

In general, x y√n
= xn−−√n ∙ y√n = xny−−−√n

II. Sometimes a given pure surd can be expressed in the form of a mixed surd.

For example,

(i) √375 = 53⋅3−−−−√

= 5√15;

(ii) ∛81 = 34−−√3

= 3∛3

(iii) ∜64 = 26−−√4
= 222−−√4= 24√4

But ∛20 can't be expressed in the form of mixed surd.

What is a Surd?

A surd is an irrational number.

We know √4 = 2, √9 = 3, √16 = 4,

But what is √2 =? and √5 =?

√2 and √5 are not rational numbers such as √4, √9 and √16 and others which are all rational numbers.

Rational numbers include terminating and non-terminating decimals.

½ is a terminating decimal, 0.5 and

1/3 is a non-terminating decimal, 0.3333….

But √2 = 1.414215…….. can be expressed as neither a terminating nor a non-terminating decimal.

So, √2 is a surd.


Examples of Surds:

2√2 or just √2 is a surd.

3√9, 3√16, 4√25, 5√100 are all numbers whose given roots are not rational numbers. They are therefore Surds.

Note:

2√8 or just √8 is irrational and therefore a Surd, but 3√8 = 2, a rational number, so, 3√8 is not a surd.

What is Not a Surd?

In the surd n√a, a is a rational number.

In n√a, if a is irrational, then n√a is not called a Surd.

From this definition:

√8 is a surd as 8 is a rational number and √8 is irrational, but
√ (2 + √3) is not a surd as (2 + √3) is not a rational number, and also
2√ (2√81) = 2√9 = 3, a rational number. Therefore, 2√ (2√81) is not a surd.

Note:

Every surd is irrational, but every irrational number is not a surd.

Order of a Surd:

In the surd n√a, n is called order of the surd.

Examples:

2√3 is a surd of order 2
4√12 is a surd of order 4,
100√3 is a surd of order 100.

Note:

If the order of surd is 2, it is optionally dropped.

2√3 is same as √3.

Types of Surds:

The following are various types of Surds:

Pure Surd:

Surds such as √3, 3√9 which are entirely irrational numbers are called pure surds


Compound Surd:

Surds such as √2 + √3, √3 - √2 are called Compound Surds.

Compound surds are sum or difference of two other surds.

Like Surds or Similar Surds:

Surds that are different multiples of same surds are called similar surds.

Example:

√80 and √45, because

√80 = √ (5 × 16) = 4√5 and

√45 = √ (5 × 9) = 3 √5

Conjugate Surds:

Two surds of the form x + √y and x - √y are called conjugate surds.

Example:

2 + √3 and 2 - √3 are called conjugate surds.

Note:

The sum or difference of two conjugate surds is a rational number.

(2 + √3) + (2 - √3) = 4, a rational number.