Simplify the square root. Assume all variables represent positive numbers. If the square root is already simplified, so state. √63 = ?

Steps to Solve

1. Find the prime factorization of 63
   To simplify $\sqrt{63}$, we begin by determining its prime factors. The prime factorization of 63 is:
   $$ 63 = 3^2 \times 7 $$

2. Rewrite the square root using the prime factors
   Using the prime factorization, we can express the square root as:
   $$ \sqrt{63} = \sqrt{3^2 \times 7} $$

3. Apply the property of square roots
   We know that $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$. Therefore, we can separate the factors:
   $$ \sqrt{3^2 \times 7} = \sqrt{3^2} \times \sqrt{7} $$

4. Simplify the square root
   Now we simplify $\sqrt{3^2}$, which equals 3:
   $$ \sqrt{3^2} = 3 $$
   So, we have:
   $$ \sqrt{63} = 3 \times \sqrt{7} $$

5. Final simplified form
   Thus, the simplified form of $\sqrt{63}$ is:
   $$ \sqrt{63} = 3\sqrt{7} $$

Conclusion

The expression is not in its simplest form initially; it simplifies to $3\sqrt{7}$.