Which of the following numbers will have an irrational square root?

Introduction / Context:
This question tests knowledge of perfect squares and the distinction between rational and irrational square roots. Recognizing perfect squares helps in many quantitative aptitude problems, especially those dealing with surds and simplification.

Given Data / Assumptions:

• We are given four integers: 1024, 2401, 4096, and 2048.
• We must decide which one does not have a perfect integer square root.
• All numbers are positive.

Concept / Approach:
A number has a rational integer square root if and only if it is a perfect square. Many powers of 2 and certain powers of primes like 7 give perfect squares. We will check whether each option is a perfect square by remembering standard squares or by expressing the number as a known power.

Step-by-Step Solution:
Step 1: Consider 1024. We know 32^2 = 1024, so 1024 is a perfect square.Step 2: Consider 2401. This is equal to 7^4 since 7^2 = 49 and 49^2 = 2401, so its square root is 49, an integer.Step 3: Consider 4096. This equals 2^12, which can be written as (2^6)^2. Hence 4096 is a perfect square with integer square root 64.Step 4: Finally consider 2048. This equals 2^11, which is not an even power of 2. There is no integer n such that n^2 = 2^11, so 2048 is not a perfect square.Step 5: Therefore the square root of 2048 cannot be an integer and will be an irrational surd.

Verification / Alternative check:
We can approximate the square root of 2048. Since 45^2 = 2025 and 46^2 = 2116, the square root of 2048 lies between 45 and 46, and is not an integer. Any non perfect square positive integer has an irrational square root, so the classification is correct.

Why Other Options Are Wrong:

• 1024: Equal to 32^2, its square root is 32, which is rational.
• 2401: Equal to 49^2, with rational square root 49.
• 4096: Equal to 64^2, so the square root is rational and integral.

Common Pitfalls:
Students sometimes confuse large powers of 2 and forget which are squares. Remember that 2^(2k) is always a perfect square, while 2^(2k+1) is not. Here 1024 = 2^10 and 4096 = 2^12, both even exponents, but 2048 = 2^11 has an odd exponent, so it is not a perfect square.

Final Answer:
The number whose square root is irrational is 2048.