Which statement about exponents of prime factors in perfect squares connects to the proof of √2?
Answer and explanation
Correct answer: In a perfect square, the exponent of 2 must be even
The relevant prime-factor principle is that every prime occurs to an even exponent in the prime factorisation of a perfect square. For example, if x = 2ᵏ times other prime factors, then x² contains 2²ᵏ, whose exponent is even. In the irrationality proof, assuming √2 = a/b in lowest terms leads to a² = 2b². Since the right side has one extra factor 2 beyond the square b², the parity of the exponent of 2 becomes impossible: the left side is a square and must have an even exponent, whereas the right side has an odd one. Hence option A states the governing fact. The other statements are false or irrelevant.
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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 relevant prime-factor principle is that every prime occurs to an even exponent in the prime factorisation of a perfect square. For example, if x = 2ᵏ times other prime factors, then x² contains 2²ᵏ, whose exponent is even. In the irrationality proof, assuming √2 = a/b in lowest terms leads to a² = 2b². Since the right side has one extra factor 2 beyond the square b², the parity of the exponent of 2 becomes impossible: the left side is a square and must have an even exponent, whereas the right side has an odd one. Hence option A states the governing fact. The other statements are false or irrelevant.
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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