Is 108 a perfect square?

Understand the Problem

The question is asking whether the number 108 can be expressed as the square of an integer. To determine this, we will check if there is an integer whose square equals 108.

Answer

No, $108$ cannot be expressed as the square of an integer.

Steps to Solve

Finding the Square Root To determine if 108 can be expressed as the square of an integer, we need to find the square root of 108. We calculate it as follows:

$$ \sqrt{108} $$

Calculation of the Square Root Now, let's simplify the square root of 108. We can start by factoring 108:

$$ 108 = 36 \times 3 $$

Since 36 is a perfect square, we can rewrite the root:

$$ \sqrt{108} = \sqrt{36 \times 3} = \sqrt{36} \times \sqrt{3} = 6\sqrt{3} $$

Determining if the Result is an Integer We see that $6\sqrt{3}$ is not an integer because $\sqrt{3}$ is an irrational number. Therefore, $6\sqrt{3}$ cannot be an integer.

No, 108 cannot be expressed as the square of an integer.

More Information

The number 108 can be factored into prime factors as $2^2 \times 3^3$. Since not all prime factors have even exponents, 108 is not a perfect square.

Tips

Overlooking Rationality: A common mistake is thinking that since $6\sqrt{3}$ is a product of integers, it must be an integer. Remember that the presence of an irrational number makes the entire expression irrational.
Forgetting Perfect Squares: Another mistake is failing to simplify the root correctly by factoring out perfect squares.

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