Simplify the square root of 63.

Step-by-step solution

1. 1

   Look for the largest perfect-square factor. Factor $63$ completely: $63=3^2\cdot 7$. The perfect squares dividing $63$ are $1$ and $9$, so the largest useful one is $9$:

   $63=9\cdot 7$

   Using the largest square in one step avoids having to simplify twice.

2. 2

   Split the radical over the product. For non-negative $a$ and $b$, $\sqrt{ab}=\sqrt a\sqrt b$:

   $\sqrt{63}=\sqrt{9\cdot 7}=\sqrt9\cdot\sqrt7$

3. 3

   Extract the exact root. $\sqrt9=3$, so

   $\sqrt{63}=3\sqrt7$

4. 4

   Confirm it is fully simplified. The remaining radicand $7$ is prime, so it contains no square factor and nothing further can come out. That is the definition of simplest radical form.

5. 5

   Check numerically. $\sqrt7=2.6457513$, so $3\sqrt7=7.9372539$; and a direct computation gives $\sqrt{63}=7.9372539$ ✓. As a rough sanity check, $63$ lies between $49=7^2$ and $64=8^2$, so the answer must sit just under $8$.