simplify radical 63

Understand the Problem

The question is asking to simplify the square root of 63, which involves breaking down the number into its prime factors to express it in a simpler form.

Answer

$3\sqrt{7}$

Steps to Solve

Prime Factorization of 63

First, we need to find the prime factors of 63. We can start by dividing by the smallest prime number.

$$ 63 = 3 \times 21 = 3 \times 3 \times 7 $$

So, the prime factorization of 63 is $3^2 \times 7$.

Applying the Square Root

Next, we apply the square root to the prime factorization. Using the property of square roots, we can separate the factors:

$$ \sqrt{63} = \sqrt{3^2 \times 7} = \sqrt{3^2} \times \sqrt{7} $$

Simplifying the Expression

Now we simplify the square root of $3^2$:

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

Thus, the expression now is:

$$ 3 \times \sqrt{7} $$

Final Simplified Form

We can conclude that the simplest form of $\sqrt{63}$ is:

$$ \sqrt{63} = 3\sqrt{7} $$

The final answer is $3\sqrt{7}$.

More Information

The square root of 63 can be simplified to $3\sqrt{7}$, which means it is expressed in its simplest radical form. This technique is commonly used in algebra when dealing with square roots and helps to make calculations easier.

Tips

Forgetting to include all prime factors when breaking down the number.
Not applying the property of square roots correctly; remember that $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$.
Failing to simplify the radical completely.

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