What is the process of writing a square root in its simplest radical form?

Answer 1

Writing a square root in its simplest radical form involves a few key steps. Here's how you can do it:

1. Identify the Square Root: Start with the expression \(\sqrt{n}\), where \(n\) is the number you're taking the square root of.

2. Factor the Number: Break down \(n\) into its prime factors or find pairs of factors. This is useful when \(n\) is not a perfect square.

3. Group the Factors: Look for pairs of the same factors. Each pair will represent a whole number when taken out of the square root.

4. Simplify: For every pair of factors, take one factor out of the square root. If \(n\) can be expressed as \(a^2 \times b\), where \(a\) is a whole number and \(b\) has no square factors, you can write: \[ \sqrt{n} = a \sqrt{b} \]

5. Final Expression: The result \(a \sqrt{b}\) is your simplest radical form. If \(b = 1\), then simply \(\sqrt{n} = a\).

Example:

Let's simplify \(\sqrt{72}\).

1. Identify the Square Root: We have \(n = 72\).

2. Factor the Number: The prime factorization of 72 is \(2^3 \times 3^2\).

3. Group the Factors: We can see there are:

   • One pair of \(3\) (from \(3^2\))
   • One pair of \(2\) (from \(2^2\)) and one leftover \(2\) (from \(2^1\))

4. Simplify:

   • Each pair gives us a factor outside the square root. The pairs give:
     • From \(3^2\), we get \(3\).
     • From \(2^2\), we get \(2\).
   • We have \(3 \times 2 = 6\) outside the square root.
   • The leftover factor inside the square root is \(2\).

5. Final Expression: Thus, \[ \sqrt{72} = 6\sqrt{2} \]

Now, \(\sqrt{72}\) is expressed in its simplest radical form as \(6\sqrt{2}\).