Real Number System

Given, √8 × √6

Need to find √8 × √6 is surd or not

⇒ we know √a × √b =

⇒ √8 × √6 can be written as

⇒

= 4√3, which is irrational number

⇒ since, 4√3 cannot be expressed as squares or cubes of any rational numbers

⇒ Hence, √8 × √6 is surd

Given,

Need to find is surd or not

⇒ we know √a × √b =

⇒ can be written as

⇒

⇒ 3 , which is irrational numbers

since, 3cannot be expressed as squares or cubes of any rational numbers

⇒ Hence, it is surd.

Given, × √5

Need to find × √5 is surd or not

⇒ we know √a × √b =

⇒ × √5 can be written as

⇒

⇒

⇒

= 2 × 3 × 5 = 30 which is not a irrational number as it can be expressed in squares form

Hence, it is not a surd

Given, 4√5 ÷ √8

Need to find 4√5 ÷ √8 is surd or not

⇒ we know √a ÷ √b =

⇒ 4√5 ÷ √8 can be written as

⇒

=

=

= is irrational number

since, cannot be expressed as squares or cubes of any rational numbers

⇒ Hence, it is surd.

Given, ∛4 ×

Need to find ∛4 × is surd or not

⇒ we know √a × √b =

⇒ ∛4 × can be written as

⇒

= 2× 2 × 2 = 8 is not irrational number as it can be expressed in cubes form

⇒ Hence, it is not a surd

Given, (10+ √3)(2 + √5)

Need to simplify it

⇒ the given expression can be written in expanded form

⇒ 20+10√5 + 2√3 +(√3 × √5)

⇒ We know √a × √b =

= 20+10√5 + 2√3 +

Hence, (10+ √3)(2 + √5) is simplified into 20+10√5 + 2√3 +

Given, (√5+√3)²

Need to simplify it

⇒ we know that (a+b)² = a²+2ab+b²

⇒ simplifying the given expression we get

⇒ (√5)²+2(√5)(√3)+ (√3)²

= 5+2+3

= 8+2

Hence, (√5+√3)² is simplified into 8+2

Given, ( – √2) ( + √2)

Need to simplify it

⇒ we know that (a–b)(a+b) = a² – b²

⇒ the given expression can be written in this form

⇒ ()² –(√2)²

= 13 –2

= 11

Hence, ( – √2) ( + √2) is simplified into 11

Given, (8+√3) (8 – √3)

Need to simplify it

⇒ we know that (a–b)(a+b) = a² – b²

⇒ the given expression can be written in this form

⇒ (8)² –(√3)²

= 64–3

= 61

Hence, (8+√3) (8 – √3) is simplified into 61

Given, 5 + 8 –

Need to simplify it

⇒ the given expression is written as follows

⇒ 5 + 8 –

= 25√3 + 48√3 – 2√3

= (25 +48 –2)√3

= 71√3

Hence, 5 + 8 – is simplified into 71√3

Given, 7∛2 + 6 –

Need to simplify it

⇒ the given expression is written as follows

⇒ 7∛2 + 6 –

= 7∛2 + 12∛2 – 3∛2

= 16∛2

Hence, 7∛2 + 6 – is simplified into 16∛2

Given, 4√72 – √50 – 7√128

Need to simplify it

⇒ the given expression is written as follows

⇒ 4 – – 7

= 24√2 –5√2 –56√2

= (24–5–56)√2

= –37√2

Hence, 4 – – 7is simplified into –37√2

Given

Need to simplify it

⇒ the given expression is written as follows

⇒ 2 + 3 – 4

⇒ 4∛5 +15∛5 – 16∛5

= (4 + 15 –16)∛5

= 3∛5

Hence, 2 + 3 – 4is simplified into 3∛5

Given,

Need to simplify it

⇒ the given number can be written as follows

⇒

= 3∛4

Hence, is simplified into 3∛4

Given,

Need to simplify it

⇒ the given number can be written as follows

⇒

= 7√2

Hence, is simplified into 7√2

Given,

Need to simplify it

⇒ the given number can be written as follows

⇒

= 8√3

Hence, is simplified into 8√3

Given,

Need to simplify it

⇒ the given number can be written as follows

⇒

= 5∛5

Hence, is simplified into 5∛5

Given, 6√5

Need to express it as pure surd

⇒ 6√5 can be expressed as (√6)² .√5

⇒

=

=

∴ is pure surd

∵ a surd with rational coefficient as unity is pure surd

Hence, 6√5 is expressed as pure surd

Given, 5∛4

Need to express it as pure surd

⇒ 5∛4 can be expressed as (∛5)³.∛4

⇒

=

=

∵ a surd with rational coefficient as unity is pure surd

∴ is a pure surd

Hence, 5∛4 is expressed as pure surd

Given, 3∜5

Need to express it as pure surd

⇒ 3∜5 can be written as

⇒

⇒

=

∵ a surd with rational coefficient as unity is pure surd

∴ is a pure surd

Hence, 3∜5 is expressed as pure surd

Given,

Need to express it as pure surd

⇒ can be expressed as follows

⇒

⇒
=

=

∵ a surd with rational coefficient as unity is pure surd

∴ is pure surd

Hence, is expressed as pure surd

Given, √5 ×

Need to simplify it

⇒ we know √a × √b =

⇒ √5 × can be written as

⇒

=

= 3

Hence, √5 × is simplified into 3

Given, ∛7 × ∛8

Need to simplify it

⇒ we know √a × √b =

⇒ ∛7 × ∛8 can be expressed as

⇒

= 2∛7

Hence, ∛7 × ∛8 is simplified into 2∛7

Given,∜8 ×

Need to simplify it

⇒ we know √a × √b =

⇒ ∜8 × can be expressed as

⇒

⇒

=

= 2∜6

Hence, ∜8 × is simplified into 2∜6

Given, ∛3 ×

Need to simplify it

⇒ we know √a × √b =

⇒ ∛3 × can be expressed as

⇒

=

=

=

=

Hence, ∛3 × is simplified into

Given, √2 or ∛3

Need to find the greater number

⇒ The order of the given irrational number is 2 and 3

⇒ now, we have to convert each irrational number into irrational number with same order

⇒ First we need to do the LCM of 2 and 3 is 6

⇒ now, each irrational number is converted into order of 6

⇒ √2 =

=

=

=

and

⇒ ∛3 =

=

=

=

⇒ is greater than

∴ ∛3 > √2

Hence, ∛3 is greater than √2

Given, ∛3 or ∜4

Need to find the greater number

⇒ The order of the given irrational number is 3 and 4

⇒ now, we have to convert each irrational number into irrational number with same order

⇒ First we need to do the LCM of 3 and 4 is 12

⇒ now, each irrational number is converted into order of 12

⇒ ∛3 =

=

=

=

And

⇒ ∜4 =

=

=

=

=

∴ is greater than

⇒ ∛3 is greater than ∜4

∴ ∛3 > ∜4

Hence, ∛3 is greater than ∜4

Given, √3 or

Need to find the greater number

⇒ The order of the given irrational number is 2 and 4

⇒ now, we have to convert each irrational number into irrational number with same order

⇒ First we need to do the LCM of 2 and 4 is 4

⇒ now, each irrational number is converted into order of 4

⇒ √3 =

=

=

=

And

⇒

∴ the greater number between and is

⇒ >

∴ > √3

Hence, is greater than √3

Given, ∜5, √3 , ∛4

Need to arrange the given numbers in ascending and descending order

⇒ The order of the given irrational number is 4, 2 and 3 respectively.

⇒ now, we have to convert each irrational number into irrational number with same order

⇒ First we need to do the LCM of 4,2 and 3 is 12

⇒ now, each irrational number is converted into order of 12

⇒ ∜5 =

=

=

=

=

And

⇒ √3 =

=

=

=

And

⇒ ∛4 =

=

=

=

∴ Ascending order is ∜5, ∛4 , √3

∴ Descending order is √3, ∛4 , ∜5

Given, ∛2, ∛4, ∜4

Need to arrange the given numbers in ascending and descending order

⇒ The order of the given irrational number is 3, 3 and 4 respectively.

⇒ now, we have to convert each irrational number into irrational number with same order

⇒ First we need to do the LCM of 3,3 and 4 is 12

⇒ now, each irrational number is converted into order of 12

⇒ ∛2 =

=

=

=

=

And

⇒ ∛4 =

=

=

=

And

⇒ ∜4 =

=

=

=

∴ Ascending order is ∛2, ∜4, ∛4

∴ Descending order is ∛4, ∜4, ∛2

Given, ∛2, ,

Need to arrange the given numbers in ascending and descending order

⇒ The order of the given irrational number is 3, 9 and 6 respectively.

⇒ now, we have to convert each irrational number into irrational number with same order

⇒ First we need to do the LCM of 3, 9 and 6 is 18

⇒ now, each irrational number is converted into order of 18

⇒ ∛2 =

=

=

=

And

⇒ =

=

=

=

And

⇒ =

=

=

=

∴ Ascending order is , , ∛2

∴ Descending order is ∛2 ,

Given, 3√2

Need to find the rationalizing factor

⇒ We know that if the product of two surds is rational then each is called a rationalizing factor of each other

⇒ 3√2 × √2 = (3)(√2)² = 6 is rational number

Hence, rationalizing factor of 3√2 is √2

Given, √7

Need to find the rationalizing factor

⇒ We know that if the product of two surds is rational then each is called a rationalizing factor of each other

⇒ √7 × √7 = 7

∴ √7 is rationalizing factor

Given,

Need to find the rationalizing factor

⇒ We know that if the product of two surds is rational then each is called a rationalizing factor of each other

⇒ can be written as

⇒

∴ × √3

⇒ (5)(3) = 15

Hence, √3 is rationalizing factor

Given, 2∛5

Need to find the rationalizing factor

⇒ We know that if the product of two surds is rational then each is called a rationalizing factor of each other

⇒ 2∛5 ×

= 2( )

= 2 ( )

= 2 (5) = 10 is rational number

Hence, is rationalizing factor

Given, 5–4√3

Need to find the rationalizing factor

⇒ We know that if the product of two surds is rational then each is called a rationalizing factor of each other

⇒ (5–4√3)(5+4√3)

⇒ 25+20√3–20√3–(16)(3)

= –23 is rational number

∴ Rationalizing factor of (5–4√3) is (5 + 4√3)

Given, √2 +√3

Need to find the rationalizing factor

⇒ We know that if the product of two surds is rational then each is called a rationalizing factor of each other

⇒ (√2 + √3)(√2 – √3)

= 4 –√6 + √6 –3

= 1 is a rational number

∴ Rationalizing factor of (√2 + √3) is (√2 – √3)

Given, √5 –√2

Need to find the rationalizing factor

⇒ We know that if the product of two surds is rational then each is called a rationalizing factor of each other

⇒ (√5 – √2)(√5 + √2)

= 25 + – – 2

= 23 is a rational number

∴ Rationalizing factor of (√5 – √2) is (√5 + √2)

Given, 2 + √3

Need to find the rationalizing factor

⇒ We know that if the product of two surds is rational then each is called a rationalizing factor of each other

⇒ (2 + √3)(2– √3)

= 4 + 2√3 –2√3 –3

= 1 is a rational number

∴ Rationalizing factor of (2+√3) is (2 – √3)

Given,

Need to rationalize the denominator

⇒ To rationalize the denominator of we must multiply the number with its denominator as follows

⇒ ×

=

=

Hence, rationalizing the denominator of we get

Given,

Need to rationalize the denominator

⇒ To rationalize the denominator of we must multiply the number with its denominator as follows

⇒ ×

=

=

=

Hence, rationalizing the denominator of we get

Given,

Need to rationalize the denominator

⇒ To rationalize the denominator of we must multiply the number with its denominator as follows

⇒ here, can be written as 2√3

⇒ ×

=

=

=

Hence, rationalizing the denominator of we get

Given,

Need to rationalize the denominator

⇒ To rationalize the denominator of we must multiply the number with its denominator as follows

⇒ ×

=

=

Hence, rationalizing the denominator of we get

Given,

Need to rationalize the denominator

⇒ To rationalize the denominator of we must multiply the number with its denominator as follows

⇒ ×

=

⇒ Now, we can write 3 as =

∴

=

∵ =

=

=

Hence, rationalizing the denominator of we get

Given,

Need to simplify by rationalizing denominator

⇒ To rationalize the denominator of we must multiply the number with its denominator as follows

⇒ ×

=

⇒ we know the denominator is in the form of a²–b² = (a+b)(a–b)

=

=

Hence, is simplified by rationalizing denominator as

Given,

Need to simplify by rationalizing denominator

⇒ To rationalize the denominator of we must multiply the number with its denominator as follows

⇒ ×

=

⇒ we know the denominator is in the form of a²–b² = (a+b)(a–b)

=

=

=

Hence, is simplified by rationalizing denominator as

Given,

Need to simplify by rationalizing denominator

⇒ To rationalize the denominator of we must multiply the number with its denominator as follows

⇒ can be written as

⇒ ×

=

⇒ we know the denominator is in the form of a²–b² = (a+b)(a–b)

=

=

Hence, is simplified by rationalizing denominator as

Given,

Need to simplify by rationalizing denominator

⇒ To rationalize the denominator of we must multiply the number with its denominator as follows

⇒ ×

=

⇒ we know that numerator is in the form of (a+b)² = a²+2ab+b²

⇒ we know the denominator is in the form of a²–b² = (a+b)(a–b)

=

=

=

=

Hence, is simplified by rationalizing denominator as

Given,

Need to simplify by rationalizing denominator

⇒ To rationalize the denominator of we must multiply the number with its denominator as follows

⇒ ×

⇒ we know the denominator is in the form of a²–b² = (a+b)(a–b)

=

=

=

=

=

Hence, is simplified by rationalizing denominator as

Given,

Need to find the values upto 3 decimal places

⇒ substitute the value of √2 1.414

⇒

= 0.707

Hence, the value of is 0.707

Given,

Need to find the values upto 3 decimal places

⇒ substitute the value of √3 1.732

⇒

= 3.46

Hence, the value of is 3.46

Given,

Need to find the values upto 3 decimal places

⇒ substitute the value of √3 1.732

⇒

= 1.887

Hence, the value of is 1.887

Given,

Need to find the values upto 3 decimal places

⇒ Substitute the value of √5 2.236, √10 3.162 and

√2 1.414

⇒

=

=

= 0.655

Hence, the value of is 0.655

Given,

Need to find the values upto 3 decimal places

⇒ Substitute the value of √5 2.236, √2 1.414

⇒

=

₌

₌

_{= 0.1022}

Hence, the value of is _0.1022

Given,

Need to find the values upto 3 decimal places

⇒ Substitute the value of √5 2.236, √2 1.414

⇒

=

=

= 4.441

Hence, the value of is 4.441

Given,

Need to find the values upto 3 decimal places

⇒ Substitute the value of √3 1.732

⇒

=

=

= 3.732

Hence, the value of is 3.732

Given,

Need to find the values upto 3 decimal places

⇒ Substitute the value of √5 2.236, √10 3.162

⇒

=

=

= 0.1854

Hence, the value of is 0.1854

Given, = a+b√6

Need to find the value of a and b

⇒ Now, we can find by rationalizing the denominator

⇒ Since, we know to rationalize the denominator of we must multiply the number with its denominator as follows

⇒ ×

=

∵ we know that numerator is in the form of (a+b)² = a²+2ab+b²

And denominator in the form of a²–b² = (a+b)(a–b)

⇒

=

= +

∴ a = and b =

Given, = a+b√3

Need to find the value of a and b

⇒ Now, we can find by rationalizing the denominator

⇒ Since, we know to rationalize the denominator of we must multiply the number with its denominator as follows

⇒ ×

=

=

=

=

=

=

=

= 7+4√ 3

∴ 7+4√3 = a+b√3

Hence, the value of a = 7 and b = 4

Given, + = a+b√5

Need to find the value of a and b

⇒ ⇒ Now, we can find by rationalizing the denominator

⇒ Since, we know to rationalize the denominator of + we must multiply the number with its denominator as follows

⇒( × )+( × )

= +

= +

= +

=

=

= 3

∴ a = 3 b = 0

Given, – = a+b√5

Need to find the value of a and b

⇒ ⇒ Now, we can find by rationalizing the denominator

⇒ Since, we know to rationalize the denominator of – we must multiply the number with its denominator as follows

⇒( × )–( × )

= –

= –

= –

=

=

⇒ = a+b√5

Hence, the value of a = 0 and b =

Given, x= 2 + √3

Need to find the values of +

⇒ By substituting the given values of x in the equation we get

⇒ x² = (2+√3)² = 4+2(2)(√3)+(3) = 7+4√3

⇒ =

=

⇒ By rationalizing method we can write as

⇒ ×

=

=

=

= 7–4√3

⇒ + = 7+4√3+7–4√3

= 14

Hence, the value of + is 14

Given, x = √3 +1

Need to find the value of

⇒ =

= ×

=

=

= √3 –1

⇒ = ((√3 +1) –(√3–1))²

= (√3+1–√3+1)²

= 4

Hence, the value of is 4

(i) 10,3

We write the given pair in the form a = bq + r, 0 r<b as follows.

10 = 3(3) + 1[3 divides 10 three time and leaves the remainder 1]

quotient = 3; remainder = 1

(ii) 5,12

We write the given pair in the form a = bq + r, 0 r<b as follows.

5 = 12(0) + 5 [12 divides 5 Zero time and leaves the remainder 5]

quotient = 0; remainder = 5

(iii) 27,3

We write the given pair in the form a = bq + r, 0 r<b as follows.

27 = 3(9) + 0 [3 divides 27 Nine time and leaves the remainder 1]

quotient = 3; remainder = 0

As we know

surd is a number in which we cant remove its square root(cube root …etc)

A. = = 2 is not surd

B. cant be simplified hence surd

C. cannot be simplified hence surd

cannot be simplified hence surd

Hence A is the answer.

On simplification

=

Hence B is the answer

On Simplification

=

=

Hence A is the answer

On rationalizing

=

=

Hence B is the answer

As we know

Prime factorization of 27 = 3

=

=

Hence C is the answer

As we know an irrational number are those number which cannot be represented in a simple fraction

A. √2 cannot be represent in simple fraction hence it is an irrational number

Hence A is true

B. √17 cannot be represent in simple fraction hence it is an irrational number

Hence B is true

C. 0.10110011100011110….cannot be represent in simple fraction hence it is a irrational number . Hence C is true

D. on simplification

=

Hence is rational number and not irrational.

Hence D is the right answer

As we know

here a is the order of surd and n is the radicand

order = 8

Radicand = 12

A is the answer

As we know

here a is the order of surd and n is the radicand

Hence C is the option

Which is B

Hence B is the answer

Option A is incorrect because √2 cannot be written in simple fraction.

hence is irrational number, hence true.

option B is incorrect as it can be written in simple form hence it is a rational number

And also √b cannot be written in simple form hence it is a irrational number

Option C

As we know surd is a number in which we cant remove its square root(cube root …etc) hence cannot represent, in simple fraction hence it is irrational number

Hence C is true

Option D

D is not true as square root of every positive integer is not always irrational

For example, square root of 4 is 2 which is rational number hence D is not true.

Hence D is the answer

As in

Option A

As we know an irrational number are those number which cannot be represented in simple fraction

Hence they are not perfect square

For example 3 is not a perfect square

also is irrational number

hence A is true

option B

index form of number means power form

as

hence B is true

Option C

Radical form means surd form

Hence C is also true

Option D

It is not true that every real number is a rational number

For example 2 is real number but not irrational number

Hence D is not true

Hence D is the right answer

using identity

(a + b)(a-b) = a² – b²

=

= (5-4)

= 1

Hence A is the answer.