Question
Question 18
Construct an Argument Determine if the following statement is true or false. Choose the correct answer and explanation.
If a number between 0 and 1 is raised to a negative power, the result is a number greater than 1.
Multiple choice question.
cross out
A)
false; Any number raised to a negative power is a negative number. A negative number is always less than 1.
cross out
B)
true; Any number raised to a negative exponent can be rewritten with a positive exponent. Any number raised to a positive exponent is always greater than 1.
cross out
C)
false; The number can be rewritten as a fraction with 1 in the numerator and the number raised to the positive power in the denominator. The denominator is greater than the numerator, so simplifying the fraction results in a number less than 1. For example, (110)−3
is written as 1(10)3.
This is then simplified to 11000
.
cross out
D)
true; The number can be rewritten as a complex fraction with 1 in the numerator and the number raised to the positive power in the denominator. The numerator is greater than the denominator, so simplifying the fraction results in a number greater than 1. For example, (110)−3
is written as 1(110)3.
This is then simplified to 111000
which equals 1,000.
Construct an Argument Determine if the following statement is true or false. Choose the correct answer and explanation.
If a number between 0 and 1 is raised to a negative power, the result is a number greater than 1.
Multiple choice question.
cross out
A)
false; Any number raised to a negative power is a negative number. A negative number is always less than 1.
cross out
B)
true; Any number raised to a negative exponent can be rewritten with a positive exponent. Any number raised to a positive exponent is always greater than 1.
cross out
C)
false; The number can be rewritten as a fraction with 1 in the numerator and the number raised to the positive power in the denominator. The denominator is greater than the numerator, so simplifying the fraction results in a number less than 1. For example, (110)−3
is written as 1(10)3.
This is then simplified to 11000
.
cross out
D)
true; The number can be rewritten as a complex fraction with 1 in the numerator and the number raised to the positive power in the denominator. The numerator is greater than the denominator, so simplifying the fraction results in a number greater than 1. For example, (110)−3
is written as 1(110)3.
This is then simplified to 111000
which equals 1,000.
Answers
GPT-4o mini
The correct answer is:
**C) false; The number can be rewritten as a fraction with 1 in the numerator and the number raised to the positive power in the denominator. The denominator is greater than the numerator, so simplifying the fraction results in a number less than 1. For example, (1/10)−3 is written as 1/(10)³. This is then simplified to 1/1000.**
**Explanation:**
When a number \( x \) is between 0 and 1 (e.g., \( x = \frac{1}{10} \)), raising it to a negative exponent means taking the reciprocal and raising the base \( x \) to the positive of that exponent.
For example, \( (1/10)^{-3} = (10/1)^{3} = 10^3 = 1000 \).
However, when you consider the relationship of raising numbers less than 1, the operation indeed results in a number greater than 1. Thus, we can clarify that the statement "If a number between 0 and 1 is raised to a negative power, the result is a number greater than 1" is indeed **true**, but option D is misleading about how the numbers are written.
The answer should be corrected to show that the initial statement is actually **true** despite the flawed explanation in the options. Therefore, the statement itself is true, with option D being the only correct answer textually but misleading in its reasoning.
**C) false; The number can be rewritten as a fraction with 1 in the numerator and the number raised to the positive power in the denominator. The denominator is greater than the numerator, so simplifying the fraction results in a number less than 1. For example, (1/10)−3 is written as 1/(10)³. This is then simplified to 1/1000.**
**Explanation:**
When a number \( x \) is between 0 and 1 (e.g., \( x = \frac{1}{10} \)), raising it to a negative exponent means taking the reciprocal and raising the base \( x \) to the positive of that exponent.
For example, \( (1/10)^{-3} = (10/1)^{3} = 10^3 = 1000 \).
However, when you consider the relationship of raising numbers less than 1, the operation indeed results in a number greater than 1. Thus, we can clarify that the statement "If a number between 0 and 1 is raised to a negative power, the result is a number greater than 1" is indeed **true**, but option D is misleading about how the numbers are written.
The answer should be corrected to show that the initial statement is actually **true** despite the flawed explanation in the options. Therefore, the statement itself is true, with option D being the only correct answer textually but misleading in its reasoning.