Question

From the following infinite list of numbers, how many are integers?

\sqrt{4096},\sqrt[3]{4096},\sqrt[4]{4096},\sqrt[5]{4096},\sqrt[6]{4096},\ldots

Answer 1

From the given infinite list of numbers, we can determine that there are 3 integers.

Step-by-step explanation:

The question asks how many numbers from the infinite list are integers. To determine this, we need to analyze the exponents of the powers in each root.

Analysis:

• For √{4096}, the exponent is 2, which is an even number, so the result is an integer.
• For ∛{4096}, the exponent is 3, which is an odd number, so the result is not an integer.
• For ∜{4096}, the exponent is 4, which is an even number, so the result is an integer.
• For ∝{4096}, the exponent is 5, which is an odd number, so the result is not an integer.
• For ∞{4096}, the exponent is 6, which is an even number, so the result is an integer.

From this analysis, we can see that only the powers with even exponents result in integers. Therefore, there are 3 integers in the infinite list of numbers.

Answer 2

Among the infinite sequence of roots of 4096, only those with exponents that are factors of 12 will result in integers. Since 4096 is 2^12, it has limited factors, hence there will be a finite number of integer roots in the infinite sequence.

Step-by-step explanation:

We are given an infinite list of numbers which are the roots of 4096. To determine how many are integers, we need to evaluate each root:

• √4096 = 4096^0.5 which equals 64, an integer.
• √[3]{4096} = 4096^(1/3) which equals 16, an integer.
• √[4]{4096} = 4096^(1/4) which equals 8, an integer.
• √[5]{4096} = 4096^(1/5) which is not an integer.
• √[6]{4096} = 4096^(1/6) which equals 4, an integer.

Since 4096 is equivalent to 2^12, we can easily determine that 4096 to the power of any factor of 12 will result in an integer. The exponents 0.5, 1/3, 1/4, and 1/6 are equivalent to the fractions 12/24, 12/36, 12/48, and 12/72, respectively, which are all factors of 12. Therefore, the number of integers in this infinite sequence will be based on the number of factors of 12. Since the factors of 12 are finite (1, 2, 3, 4, 6, 12), there will only be a finite number of exponents that produce an integer when 4096 is raised to them. Thus, out of the list provided and extending indefinitely, there will be a finite number of integers.