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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.
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.