Question

QuestionProve that the sum of 34 and 10−−√ is irrational. Show your work, and in 1–2 sentences, explain why the sum is an irrational number.(2 points)

Answer 1

Proof:
First show √10 is irrational. Suppose √10 = a/b with integers a,b in lowest terms (gcd(a,b)=1). Then a^2 = 10 b^2, so a^2 is even and hence a is even. Write a = 2k; then 4k^2 = 10 b^2, so 2k^2 = 5 b^2. Thus b^2 (and hence b) is even, contradicting gcd(a,b)=1. Therefore √10 is irrational.

Now assume 34 + √10 were rational. Subtracting the rational number 34 would give √10 = (34+√10) − 34 rational, contradicting that √10 is irrational. Hence 34 + √10 is irrational.

Explanation (1–2 sentences):
√10 is irrational, and adding a rational number (34) to an irrational number cannot produce a rational number; if the sum were rational, subtracting 34 would make √10 rational, which is impossible.