Carcass wrote:

The integer v is greater than 1. If v is the square of an integer, which of the following numbers must also be the square of an integer?

Indicate all such numbers.

A) 81v

B) 25v + 10√v + 1

C) 4v² + 4√v + 1

We're told that

v is the square of an integerLet's say v = k², where k is a positive integer

This means that

√v = kNow let's examine the answer choice....

A) 81v = (81)(k²)

= (9)(9)(k)(k)

= (9k)(9k)

= (9k)²

Since k is an integer, we can be certain that 9k is an integer

So, 81v IS the square of an integer

B) 25v + 10√v + 1 = 25k² + 10

k + 1

= (5k + 1)(5k + 1)

= (5k + 1)²

Since k is an integer, we can be certain that 5k is an integer, and if 5k is an integer, we can be certain that 5k + 1

So, 25v + 10√v + 1 IS the square of an integer

C) 4v² + 4√v + 1 = 4(k²)² + 4

k + 1

= 4k⁴ + 4

k + 1

This expression cannot be factored into the form (some integer)²

To verify this, let's replace k with some integer and see what we get.

Let k =

2 If k =

2, then 4k⁴ + 4k + 1 = 4

(2)⁴ + 4

(2) + 1

= 64 + 8 + 1

= 73

Since 73 is NOT the square of an integer, we CANNOT conclude that 25v + 10√v + 1 is the square of an integer

Answer: A and B

Cheers,

Brent

_________________

Brent Hanneson – Creator of greenlighttestprep.com

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