The following numbers are not perfect squares. Give a reason.
a) 1037
b) 23453
c) 222000
d) 506070
e) 16000
f) 222222
g) 7928
h) 89722
i) 9000
Answer (2)
None of the listed numbers are perfect squares because their prime factorizations contain prime factors raised to an odd exponent. This rule applies to all of the examined numbers, confirming they cannot be expressed as the square of an integer. Each factorization reveals at least one prime with an odd exponent, which prevents the number from being a perfect square.
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To determine whether a number is not a perfect square, we need to understand what a perfect square is. A perfect square is an integer that is the square of an integer. In other words, if there is an integer n such that n 2 equals the given number, then that number is a perfect square. Let's examine each number to see why they are not perfect squares:
a) 1037
The square root of 1037 is approximately 32.2. Since 32.2 is not an integer, 1037 is not a perfect square.
b) 23453
The square root of 23453 is approximately 153.16. Since 153.16 is not an integer, 23453 is not a perfect square.
c) 222000
The square root of 222000 is approximately 471.5. Since 471.5 is not an integer, 222000 is not a perfect square.
d) 506070
The square root of 506070 is approximately 711.34. Since 711.34 is not an integer, 506070 is not a perfect square.
e) 16000
Thinking about the square of integers near 16000, the square root is approximately 126.49. Since 126.49 is not an integer, 16000 is not a perfect square.
f) 222222
The square root of 222222 is approximately 471.4. Since 471.4 is not an integer, 222222 is not a perfect square.
g) 7928
The square root of 7928 is approximately 89.03. Since 89.03 is not an integer, 7928 is not a perfect square.
h) 89722
The square root of 89722 is approximately 299.536. Since 299.536 is not an integer, 89722 is not a perfect square.
i) 9000
The square root of 9000 is approximately 94.87. Since 94.87 is not an integer, 9000 is not a perfect square.
In general, to determine whether a number is a perfect square, you can take the square root of that number and check if the result is an integer. If it is not, then the number is not a perfect square.
