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Is 164 a perfect square?

A. Yes, because [tex]$4 \cdot 41 = 164$[/tex], and 4 is a perfect square, so 164 is also a perfect square.

B. Yes, because [tex]$164 = 100 + 64$[/tex], and 100 and 64 are both perfect squares, so 164 will also be a perfect square.

C. Yes, because perfect squares must be even numbers and 164 is an even number.

D. No, because there is no whole number that when multiplied by itself gives 164.

Answer :

To determine if a number is a perfect square, we need to check if there exists an integer [tex]$k$[/tex] such that

[tex]$$
k^2 = 164.
$$[/tex]

Here are the steps:

1. First, calculate the square root of 164:

[tex]$$
\sqrt{164} \approx 12.8062.
$$[/tex]

2. Next, we take the integer part of this square root, which is 12. This means we consider [tex]$12^2$[/tex]:

[tex]$$
12^2 = 144.
$$[/tex]

3. Since 144 is less than 164, we can also check the next integer, 13:

[tex]$$
13^2 = 169.
$$[/tex]

4. Notice that

[tex]$$
144 < 164 < 169.
$$[/tex]

This shows that there is no integer [tex]$k$[/tex] such that

[tex]$$
k^2 = 164.
$$[/tex]

Therefore, 164 is not a perfect square.

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