Prove that (root 2 + root 5 ) is irrational.

Rational numbers are integers that are expressed in the form of p / q where p and q are both co-prime numbers and q is non zero. Irrational numbers do not satisfy this condition.

Answer: Hence, it is proved that √2 + √5 is an irrational number.

Let's find if √2 + √5 is irrational.

Explanation:

To prove that √2 + √5 is an irrational number, we will use the contradiction method.

Let us assume that √2 + √5 is a rational number with p and q as co-prime integers and q ≠ 0

⇒ √2 + √5 = p/ q

Squaring both sides:

⇒ 7 + 2√10 = p²/q²

⇒ √10 = (p²/q² - 7) / 2

⇒ We know that (p²/q² - 7) / 2 is a rational number.

Also, we know √10 = 3.1622776... which is irrational.

Since we know that √10 is an irrational number, but an irrational number cannot be equal to a rational number.

This leads to a contradiction that √2 + √5 is a rational number.

Thus, √2 + √5 is an irrational number by contradiction method.