Question

12. Prove that 3 + 2 root 5 is irrational.
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Answer 1

Step-by-step explanation:

Let us assume that 3 + 2√5 is a rational number. This shows (a-3b)/2b is a rational number. But we know that √5 is an irrational number. So, it contradicts our assumption

Answer 2

Step-by-step explanation:

Given:-

3+2√5

To find:-

Prove that 3 + 2 √5 is an irrational number?

Solution:-

Let us assume that 3 + 2 √5 is a rational number

It must be in the form of p/q

Where p and q are integers and q≠0

Let 3+2√5 = a/b

Where a and b are co primes

=> 2√5 = (a/b)-3

=>2√5 = (a-3b)/b

=>√5 = (a-3b)/(2×b)

=>√5 = (a-3b)/2b

=>√5 is in the form of p/q

=>√5 is a rational number

But √5 is not a rational number

√5 is an irrational number

This contradicts to our assumption that 3+2√5 is a rational number.

So Our assumption is wrong

3+2√5 is not a rational number

3+2√5 is an irrational number.

Hence , Proved

Answer:-

3+2√5 is an irrational number.

Note:-

The sum of a rational number and an irrational number is an irrational number

3 is a rational number

2√5 is an irrational number

Their sum 3+2√5 is an irrational number.

Used formulae:-

• The numbers in the form of p/q where p and q are integers and q≠0 are called rational numbers.

• The numbers are not in the form of p/q where p and q are integers and q≠0 are called irrational numbers.

• √2,√3,√5...,π... are irrational numbers.

• The sum of a rational number and an irrational number is an irrational number