On simplification, {(2-1)}-1 gives a number. Consider the following statements…
2019
On simplification, {(2-1)}-1 gives a number. Consider the following statements about that number:
A. Prime number
B. Even number
C. Multiple of 2
D. Odd number
Which combination of the above statements is correct?
Answer: A. A, B, C — Concept. For a non-zero base and integer exponents, a negative exponent means a reciprocal: a-n = 1/an. A power raised to a further power multiplies the two…
- A.
A, B, C
- B.
A, C, D
- C.
B, C, D
- D.
A, B, D
Attempted by 4 students.
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Correct answer: A
Concept. For a non-zero base and integer exponents, a negative exponent means a reciprocal: a-n = 1/an.
A power raised to a further power multiplies the two exponents: (am)n = amn.
Applying the exponent -1 twice therefore brings the base back unchanged: (a-1)-1 = a(-1) x (-1) = a1 = a.
For classifying a whole number: it is even, equivalently a multiple of 2, when 2 divides it exactly; it is odd when 2 does not divide it exactly; and it is prime when it is greater than 1 and has exactly two positive divisors.
Application. Simplify the given expression one step at a time:
Inner power: 2-1 = 1/21 = 1/2.
Outer power: (1/2)-1 = 1 ÷ (1/2) = 2.
The same result straight from the power rule: {(2-1)}-1 = 2(-1) x (-1) = 21 = 2.
So the simplified value is 2.
Contrast. Now test each of the four statements on the value 2:
Statement | Test on 2 | Holds? |
|---|---|---|
A. Prime number | The positive divisors of 2 are 1 and 2, which is exactly two divisors | Yes |
B. Even number | 2 = 2 x 1, so 2 is exactly divisible by 2 | Yes |
C. Multiple of 2 | 2 = 2 x 1, so 2 is the first multiple of 2 | Yes |
D. Odd number | 2 is exactly divisible by 2, so it is not of the form 2k + 1 | No |
Result. The statements that hold for the value 2 are A (prime number), B (even number) and C (multiple of 2); D (odd number) does not hold. The correct combination is therefore A, B, C. Note that 2 is the smallest prime and the only even prime, which is why the prime statement and the even statement can hold together.