Simplifying Surds

 

What Does it Mean to Simplify a Surd?

• Simplifying a surd means breaking down the number under the square root  into its prime factors, looking for perfect squares, and then taking those perfect squares outside the root.
• This process often makes the surd more concise and easier to work with, especially when adding, subtracting, or comparing surds.

Simplifying surds makes them easier to understand and work with. For example, instead of dealing with $\sqrt{50}$, we can simplify it to $5\sqrt{2}$, which is much cleaner

Steps to Simplify Surds

1. Find Factors of the Number: Look for factors that include a perfect square (like 4, 9, 16).
2. Separate into Two Roots: Split the number under the square root into two parts.
3. Simplify the Perfect Square: Take the square root of the perfect square part.

Example: Simplifying $\sqrt{50}$

1. Break 50 into factors: $500 = 25 \times 2$
2. Write as two square roots: $$\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2}$$
3. Simplify $\sqrt{25}$ to 5: $$= 5\sqrt{2}$$

So, $\sqrt{50}$ simplifies to $5\sqrt{2}$.