check whether the following are rational numbers or not (√2+ √3²)
Answers
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.
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Similar questions :
Give an example of each, of two irrational numbers whose:
(i) Difference is a rational number.
(ii) Difference is an irrational number.
(iii) Sum is a rational number.
(iv) Sum is an irrational number.
(v) Product is a rational number.
(vi) Product is an irrational number.
(vii) Quotient is a rational number.
(viii) Quotient is an irrational number.
brainly.in/question/15896923
Identify the following as rational or irrational numbers. Give the decimal representation of rational numbers:
(i)√4 (ii) 3√18 (iii) √1.44 (iv) √(9/27)(v) -√64 (vi) √100
brainly.in/question/15896924
(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
0 is a rational number
(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.
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