9.3 Add and Subtract Square Roots

Operations with Square Roots

Adding and subtracting square roots works a lot like combining like terms with variables. Just as you can combine $3x + 5x$ into $8x$, you can combine square roots that share the same radicand. The key skill here is recognizing when two radicals are "like" and when you need to simplify first before combining.

Adding and Subtracting Like Square Roots

Like square roots have the same radicand (the number under the radical sign). You can only add or subtract square roots when their radicands match.

To combine them, add or subtract the coefficients (the numbers in front of the radical) and keep the radicand the same:

• $2\sqrt{5} + 3\sqrt{5} = 5\sqrt{5}$
• $7\sqrt{3} - 4\sqrt{3} = 3\sqrt{3}$

If no coefficient is written, it's 1. So $\sqrt{7} + \sqrt{7} = 1\sqrt{7} + 1\sqrt{7} = 2\sqrt{7}$.

You cannot combine unlike square roots. For example, $2\sqrt{3} + 5\sqrt{7}$ stays exactly as it is. There's no way to simplify further because the radicands (3 and 7) don't match.

Simplifying Radical Expressions

Sometimes two radicals don't look alike at first, but they become like terms after you simplify. That's why simplifying always comes before combining.

To simplify a square root, factor out the largest perfect square from the radicand:

1. Find the largest perfect square factor of the radicand
2. Rewrite the radicand as a product of that perfect square and the remaining factor
3. Take the square root of the perfect square and move it to the coefficient

If there's already a coefficient, multiply it by whatever comes out of the radical:

Combining Radicals That Need Simplifying First

This is where the two skills come together. When you see an expression like $\sqrt{12} + \sqrt{27}$, the radicands don't match yet. But after simplifying each term, they might.

Steps for adding/subtracting radicals:

1. Simplify each radical separately
2. Identify like terms (matching radicands)
3. Combine the coefficients of like terms

A common mistake is trying to combine radicals before simplifying. Always simplify first, then check if the radicands match.

Rational vs. Irrational Numbers

• Rational numbers can be written as a fraction of two integers (like $\frac{3}{4}$ or $-2$)
• Irrational numbers can't be written as a fraction. Their decimals go on forever without repeating

Square roots of perfect squares (like $\sqrt{9} = 3$ or $\sqrt{25} = 5$) are rational. Square roots of non-perfect squares (like $\sqrt{2}$ or $\sqrt{7}$) are irrational. You can't write $\sqrt{2}$ as an exact decimal or fraction, which is why we leave it in radical form.

The Distributive Property with Square Roots

The distributive property works the same way with radicals as it does with variables. Multiply the outside term by each term inside the parentheses:

$3(\sqrt{2} + \sqrt{5}) = 3\sqrt{2} + 3\sqrt{5}$

Notice that $3\sqrt{2}$ and $3\sqrt{5}$ have different radicands, so they can't be combined further. The expression is already fully simplified.