Math, asked by aniketmohanpuri8428, 4 hours ago

check whether the following are rational numbers or not (√2+ √3²)

Answers

Answered by gyaneshwarsingh882
0

Answer:

                                                       

Given : (i) √7  = 2.6457513111…..

Since the number is non-terminating non-recurring decimal expansion.

Hence, it is an irrational number.

 

(ii) √4 = 2 A perfect square root of 2.  

Since the number is a rational number as it can represented in p/q (2/1) form.

 

(iii) 2 + √3

Here, 2 is a rational number and √3 is an irrational number

We know that, the sum of a rational and irrational number is an irrational number.

Hence , 2 + √3 is an irrational number.

 

(iv) √3 + √2

Here, √3 is a irrational number and √2 is an irrational number.

We know that, the sum of two irrational numbers  is an irrational number.

Hence , √3 + √2 is an irrational number.

 

(v) √3 + √5

Here, √3 is a irrational number and √5 is an irrational number.

We know that, the sum of two irrational numbers  is an irrational number.

Hence , √3 + √5 is an irrational number.

 

(vi) (√2 – 2)²

(√2 – 2)² = √2² + 2² –2 × √2 × 2

[(a - b)² = a² + b² - 2ab]

= 2 + 4 - 4√2

= 6 - 4 √2

Here, 6 is a rational number and  4√2 is an irrational number.

We know that, the sum of a rational and irrational number is an irrational number.

Hence, (√2 – 2)²  is an irrational number.

 

(vii) (2 – √2)(2 + √2)

(2 – √2)(2 + √2) =  2² − √2²

[(a + b)(a – b) = a² – b²]

= 4 – 2  

= 2 or 2/1

Since, 2 is a rational number.

Hence, (2 – √2)(2 + √2) is a rational number.

 

(viii) (√3 + √2)²

= (√3)² + (√2)² + 2√3 x √2

[(a - b)² = a² + b² - 2ab]

= 3 + 2 + 2√6

= 5 + 2√6

Since, the sum of a rational number and an irrational number is an irrational number.

Hence, (√3 + √2)² is an irrational number.

 

(ix) √5 – 2

√5 is an irrational number and 2 is a rational number.

The difference of an irrational number and a rational number is an irrational number.

Hence,, √5 – 2 is an irrational number.

 

(x) √23 = 4.79583152331…

Since the number is non-terminating non-recurring decimal expansion.

Hence, it is an irrational number.

 

(xi) √225 = 15 = 15/1

Since the number is rational number as it can represented in p/qform.

 

 

(xii) 0.3796

Since the number has terminating decimal expansion.

Hence, it is an rational number.

 

(xiii) 7.478478….. = 7.478

478 is repeating continuously so it has non terminating repeating (recurring ) decimal expansion

Hence, it is a rational number.

 

(xiv) 1.101001000100001…

Since the number is non-terminating non-repeating.

Hence, it is an irrational number.

HOPE THIS ANSWER WILL HELP YOU…..

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(i) Let two irrational numbers be √2 and √2 .

Now, difference of two irrational numbers be :  

(√2) – (√2) = 0

0 is the rational number.

 

(ii) Let two irrational numbers are 4√3 and 2√3 .

Now, difference of two irrational numbers be :  

(4√3) – (2√3) = 2√3

2√3 is an irrational number.

 

(iii) Let two irrational numbers be √5 and - √5 .

Now, sum of two irrational numbers be :  

(√5) + (-√5) = 0

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(iv) Let two irrational numbers be 2√5 and 3√5 .

Now, sum of two irrational numbers be :  

(2√5) + (3√5) = 5√5

5√5 is an irrational number.

 

(v) Let two irrational numbers be √8 and √2 .

Now, product of two irrational numbers be :  

√8 ×  √2 = √16 = 4

4 is a rational number.

 

(vi) Let two irrational numbers be  √2 and √3 .

Now, product of two irrational numbers be :  

√2 × √3 = √6

√6 is an irrational number.

 

(vii) Let two irrational numbers be √8 and √2 .

Now, quotient of two irrational numbers be :  

√8 / √2 = √8/2 = √4 = 2

2 is a rational number.

 

(viii) Let two irrational numbers are √2 and √3 .

Now, quotient of two irrational numbers be :  

√2 / √3 = √2 × √3 /√3 ×√3

= √6/3

√6/3 is an irrational number.

 

HOPE THIS ANSWER WILL HELP YOU…..

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