So, since we know 3 is a rational number, 3√2 is irrational (3 = 3/1; 1 ≠ 0)
Now we know that 3√2 is irrational.
Theorem: The sum of a rational number with an irrational number is irrational.
So, since we know 5 is a rational number, 5 + 3√2 is irrational (5 = 5/1; 1 ≠ 0)
Thus we proved that 5 + 3√2 is an irrational number.

" /> So, since we know 3 is a rational number, 3√2 is irrational (3 = 3/1; 1 ≠ 0)
Now we know that 3√2 is irrational.
Theorem: The sum of a rational number with an irrational number is irrational.
So, since we know 5 is a rational number, 5 + 3√2 is irrational (5 = 5/1; 1 ≠ 0)
Thus we proved that 5 + 3√2 is an irrational number.

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Question:

Given that √2 is irrational, prove that (5+3√2) is an irrational number.

Solution:

Given that √2 is irrational.
We know that the theorem "The product of any irrational number with a rational number is irrational".
So, since we know 3 is a rational number, 3√2 is irrational (3 = 3/1; 1 ≠ 0)
Now we know that 3√2 is irrational.
Theorem: The sum of a rational number with an irrational number is irrational.
So, since we know 5 is a rational number, 5 + 3√2 is irrational (5 = 5/1; 1 ≠ 0)
Thus we proved that 5 + 3√2 is an irrational number.