Question

prove that 5 + 2 √3 is an irrational number​

Answer 1

Answer:

Let assume that 5 +2 √3 be a rational number

5 + 2√3 = a /b {where a and b are co-prime}

2√3 = a / b + 5

√3 =(a + 5b)/2b

since a/ b is a rational no.

therefore ( a +5b)/2 b js also rational no.

therefore √3 is also rational

since it contradicts the fact that √3 is irrational

therefore our assumption is wrong

therefore 5 + 2√3 is an irrational number

Hence proved!!!!!

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Answer 2

Answer:

Step-by-step explanation:

Let us assume that 5+2√3 is a rational number which can be expressed in the form p/q where p and q are rational numbers and q≠0

∴ 5+2√3 = p/q

2√3 = p/q - 5

2√3 = p-5q/q

√3 = p-5q/2q

On squaring both sides,

3 = p² + 25q² - 10pq/4q²

From this,

3 is a factor of p² +25q² +10pq/4q²

∴√3 is a factor of p/q

∴√3 is a rational number.

But this contradicts the fact that √3 is an irrational number.

Therefore, our assumption is wrong

∴5+2√3 is an irrational number

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