radical 108 simplified

Understand the Problem

The question is asking to simplify the square root of 108. This involves breaking down 108 into its prime factors and finding the square root of perfect squares within it.

Answer

$6\sqrt{3}$

Steps to Solve

Factor the number 108

To simplify the square root, we first find the prime factorization of 108.

We can divide 108 by 2:

$$ 108 \div 2 = 54 $$

Next, we factor 54:

$$ 54 \div 2 = 27 $$

Now, we factor 27:

$$ 27 \div 3 = 9 $$

And we factor 9:

$$ 9 \div 3 = 3 $$

So, the prime factors of 108 are:

$$ 108 = 2^2 \times 3^3 $$

Identify perfect squares

Now, we look for the perfect squares in the prime factors. We have:

$2^2$ (which is a perfect square)
$3^2$ from $3^3$ (we can take out one 3)

Take the square root

Now, we can simplify the square root using the perfect squares identified:

$$ \sqrt{108} = \sqrt{2^2 \times 3^2 \times 3} $$

This can be broken down into:

$$ \sqrt{2^2} \times \sqrt{3^2} \times \sqrt{3} $$

Calculating the square roots gives us:

$$ 2 \times 3 \times \sqrt{3} $$

Combine the results

Finally, we combine the results:

$$ 2 \times 3 = 6 $$

So,

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

The simplified form of the square root of 108 is $6\sqrt{3}$.

More Information

The square root of 108 simplifies to $6\sqrt{3}$, which means that the number contains a factor of 3 that cannot be simplified into a whole number. This technique of breaking down numbers into prime factors is useful for simplifying square roots and understanding their components.

Tips

Failing to fully factor the number would result in an incorrect simplification.
Forgetting to include all perfect squares in the factorization can lead to missing terms in the final answer.

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