Using exponents of prime factors in perfect squares, which idea is correct in the proof of √2?
Answer and explanation
Correct answer: In a perfect square, the exponent of 2 must be even
The fundamental exponent rule says that when a number is squared, every prime exponent in its factorisation is doubled. Therefore each prime, including 2, occurs to an even exponent in a perfect square. In the proof, assume √2 = r/s in lowest terms. Squaring gives r² = 2s². The left side is a perfect square and must contain an even exponent of 2, while the right side contains the factor 2 multiplied by the square s², producing an odd exponent of 2 relative to the square structure. This incompatibility leads to the conclusion that the original rational assumption is impossible. Thus option A is correct; the other choices confuse the rule with unrelated or false claims.
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What is the correct answer to this question?
In a perfect square, the exponent of 2 must be even
Why is this the correct answer?
The fundamental exponent rule says that when a number is squared, every prime exponent in its factorisation is doubled. Therefore each prime, including 2, occurs to an even exponent in a perfect square. In the proof, assume √2 = r/s in lowest terms. Squaring gives r² = 2s². The left side is a perfect square and must contain an even exponent of 2, while the right side contains the factor 2 multiplied by the square s², producing an odd exponent of 2 relative to the square structure. This incompatibility leads to the conclusion that the original rational assumption is impossible. Thus option A is correct; the other choices confuse the rule with unrelated or false claims.
Which subject and chapter does this question cover?
This is a Class 9 Mathematics question. Chapter: Number Systems. Topic: Proof of irrationality of square root 2 and square root 3.
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