UpStudy Free Solution:

To simplify \(\sqrt { 63} \), we need to find the prime factorization of 63 and then simplify the square root.

First, find the prime factorization of 63:

\[63 = 3 \times 21\]

\[21 = 3 \times 7\]

So, the prime factorization of 63 is:

\[63 = 3 \times 3 \times 7\]

Now, we can simplify the square root:

\[\sqrt { 63} = \sqrt { 3 \times 3 \times 7} \]

Since \(\sqrt { a \times a} = a\):

\[\sqrt { 63} = \sqrt { 3^ 2 \times 7} = 3\sqrt { 7} \]

So, the simplified form of \(\sqrt { 63} \) is:

\[\sqrt { 63} = 3\sqrt { 7} \]

This is the exact answer using radicals.

Supplemental Knowledge

Simplifying radicals involves expressing a square root in its simplest form. This often requires breaking down the number inside the square root into its prime factors and then simplifying based on the properties of square roots.

The steps to simplify a square root are as follows:

1. Prime Factorization: Break down the number inside the square root into its prime factors.

2. Pairing Factors: Identify pairs of prime factors since \(\sqrt { a \times a} = a\).

3. Simplify: Move each pair of factors outside the square root, leaving any unpaired factors inside.

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