9.2 Simplify Square Roots

Properties of Square Roots

The product property and quotient property of square roots are the two main tools you'll use to simplify radicals. Once you know how to apply them, simplifying square roots becomes a repeatable process rather than guesswork.

Product Property of Square Roots

The product property states:

$\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$

This means you can split a square root of a product into the product of two separate square roots. The whole strategy for simplifying square roots comes down to this: find the largest perfect square factor hiding inside the number under the radical, then pull it out.

Perfect squares are numbers that result from squaring a whole number: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, and so on.

Here's the step-by-step process for simplifying a square root:

1. Look at the number under the radical and find its largest perfect square factor.
2. Rewrite the number as a product of that perfect square and whatever is left over.
3. Split the radical using the product property.
4. Take the square root of the perfect square factor.

Example: Simplify $\sqrt{50}$

1. The largest perfect square factor of 50 is 25 (since $25 \times 2 = 50$).
2. Rewrite: $\sqrt{50} = \sqrt{25 \cdot 2}$
3. Split: $\sqrt{25} \cdot \sqrt{2}$
4. Simplify: $5\sqrt{2}$

Another example: Simplify $\sqrt{72}$

1. The largest perfect square factor of 72 is 36 (since $36 \times 2 = 72$).
2. Rewrite: $\sqrt{72} = \sqrt{36 \cdot 2}$
3. Split: $\sqrt{36} \cdot \sqrt{2}$
4. Simplify: $6\sqrt{2}$

A common mistake is not finding the largest perfect square factor. For instance, you could write $\sqrt{72} = \sqrt{4 \cdot 18} = 2\sqrt{18}$, which is correct but not fully simplified. You'd then need to simplify $\sqrt{18}$ again. Using the largest perfect square factor saves you that extra step.

Quotient Property of Square Roots

The quotient property states:

$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}, \text{ where } b \neq 0$

This lets you split a square root of a fraction into a fraction of two separate square roots. From there, you simplify the numerator and denominator individually.

Example: Simplify $\sqrt{\frac{36}{49}}$

Both 36 and 49 are perfect squares, so this one is straightforward:

$\sqrt{\frac{36}{49}} = \frac{\sqrt{36}}{\sqrt{49}} = \frac{6}{7}$

Example with a non-perfect-square numerator: Simplify $\sqrt{\frac{18}{25}}$

1. Split the fraction: $\frac{\sqrt{18}}{\sqrt{25}}$
2. Simplify the denominator: $\sqrt{25} = 5$
3. Simplify the numerator using the product property: $\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}$
4. Result: $\frac{3\sqrt{2}}{5}$

Order of Operations with Square Roots

When square roots appear in larger expressions, follow the standard order of operations (PEMDAS). The key rule: simplify each square root first, then carry out the remaining operations.

1. Parentheses first: simplify anything inside grouping symbols.
2. Exponents and roots: evaluate square roots at this stage.
3. Multiplication and Division: left to right.
4. Addition and Subtraction: left to right.

Example: Simplify $2 + \sqrt{18} - 3 \cdot \sqrt{4}$

1. Simplify the square roots: $\sqrt{18} = 3\sqrt{2}$ and $\sqrt{4} = 2$

2. Substitute back: $2 + 3\sqrt{2} - 3 \cdot 2$

3. Multiply: $2 + 3\sqrt{2} - 6$

4. Combine the constant terms: $-4 + 3\sqrt{2}$

After simplifying, combine like radical terms if possible. Like radical terms have the same number under the radical sign. For example:

$2\sqrt{3} + 5\sqrt{3} = 7\sqrt{3}$

But $2\sqrt{3} + 5\sqrt{2}$ cannot be combined because the radicands (3 and 2) are different.

Additional Concepts

• The square root function $f(x) = \sqrt{x}$ is the inverse of the squaring function $f(x) = x^2$ (for non-negative values of $x$).
• Surds are square roots (or other roots) that result in irrational numbers and can't be simplified to a whole number. For example, $\sqrt{2}$ is a surd, but $\sqrt{9}$ is not.
• Conjugates are used to rationalize denominators that contain square roots. The conjugate of $\sqrt{2} + 1$ is $\sqrt{2} - 1$. Multiplying a denominator by its conjugate eliminates the radical because $(a + b)(a - b) = a^2 - b^2$.