Question

3√5*2√5 is an irrational number. (true/false)​

Answer 1

Let us assume 3√2 + √5 = a rational number = p / q  where p and q are integers and q is not 0.  

Let us assume that p / q is in reduced form and have no common factors and are prime to each other.  

 (3 √2 + √5 )² = p² / q²           18 + 5 + 6 √10 = p² / q²      √10 = (p²-23 q²) / 6 q²  =  m/n where m and n are integers and n≠ 0

0 = m² / n²       or,     m * m = 2 * 5 * n * n       Since m and n are prime to each other, m must have a factor of 2 and a factor of 5 also.  Let   m = 2 * 5* k.        

2 * 5 * k * 2 * 5 * k = 2 * 5 * n * n            

2 * 5 * k * k = n * n

Hence n must have as a factor 2 and 5 also.  So, n = 2 * 5 * h.  We started with p/q where they are co-prime and derived that they have common factors 2 and 5.   Our assumption that the given number is a rational number , must be wrong. Hence, 3 √2 + √5  is irrational.

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we know that (a +b)(a - b) = a² - b²(3√2 + √5) (3√2 - √5) = (3√2)² - (√5)²  = 9*2 - 5 = 13. So the product of the two irrational numbers is rational and in fact, is a positive integer.

Answer 2

Answer: False, 3√5*2√5 is not an irrational number.

Step-by-step explanation:

Since we have given that

$3\sqrt{5}\times 2\sqrt{5}\\\\=3\times 2\times \sqrt{5}\times \sqrt{5}\\\\=6\times 5\\\\=30$

Since we know that 30 is a rational number as we can write in p/q form.

Hence, False, 3√5*2√5 is not an irrational number.

# learn more:

State true or false 1)all natural numbers are rational number 2)-5 is an irrational number 3) product of two irrational numbers is always irrational number 4)all the rational number are integer​

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