Lesson 3: Arithmetic with Square Roots

Property

To Simplify a Square Root:

1. Factor any perfect squares from the radicand.
2. Use the product rule to write the radical as a product of two square roots.
3. Simplify the square root of the perfect square.

Examples

• To simplify $\sqrt{50}$, we find the perfect square factor 25. So, $\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25}\sqrt{2} = 5\sqrt{2}$.
• To simplify $\sqrt{72}$, we use the largest perfect square factor, 36. So, $\sqrt{72} = \sqrt{36 \cdot 2} = \sqrt{36}\sqrt{2} = 6\sqrt{2}$.
• To simplify $\sqrt{108}$, we factor it as $\sqrt{36 \cdot 3}$. This simplifies to $\sqrt{36}\sqrt{3} = 6\sqrt{3}$.

Explanation

Simplifying a radical means pulling out any perfect square factors hiding inside the radicand. Find the largest perfect square that divides your number, separate it, and take its root, leaving the rest inside.

Property

Square roots with identical radicands are called like radicals. We can add or subtract like radicals in the same way that we add or subtract like terms, namely by adding or subtracting their coefficients. For example, $2\sqrt{2} + 3\sqrt{2} = 5\sqrt{2}$.

Examples

• Combining like radicals: $9\sqrt{5} - 4\sqrt{5} = (9-4)\sqrt{5} = 5\sqrt{5}$.
• Unlike radicals cannot be combined: $8\sqrt{3} + 2\sqrt{7}$ cannot be simplified into a single term.
• Sometimes you must simplify first: $\sqrt{18} + \sqrt{50} = \sqrt{9 \cdot 2} + \sqrt{25 \cdot 2} = 3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}$.

Explanation

You can only add or subtract radicals if the number inside the square root (the radicand) is exactly the same. Think of $\sqrt{3}$ as a variable like $x$. You can combine $5x + 2x$ but not $5x + 2y$.